Activity 1

Count the Dots—Binary Numbers

Summary

Computers store and send information using only two symbols: zero and one. How can we use just these two symbols to stand for words and numbers? This activity will show you how!

Curriculum Links

• Math: Numbers — Learning about number systems that aren't base 10 (our normal counting system). Learning to write numbers in base two.
• Math: Patterns — Continuing a pattern and figuring out the rule behind it. Looking at patterns in powers of two (numbers like 2, 4, 8, 16...).

Skills

• Counting
• Matching
• Sequencing (putting things in order)

Ages

7 and up

Materials

You will need a set of five binary cards (see page 6) to show the class. Big cards with sticker dots work well for this.

Each child will need:

• A set of five cards, cut out from the Photocopy Master on page 6
• Worksheet: Binary Numbers (page 5)

There are also extra activities you can try, using these worksheets:

• Working with Binary (page 7)
• Sending Secret Messages (page 8)
• Fax Machines and Modems (page 9)
• Counting Higher Than 31 (page 10)
• More on Binary Numbers (page 11)

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Binary Numbers

Introduction

Before handing out the worksheet on page 5, it helps to show the class how this works first.

For this activity, you'll need five cards with dots on one side and nothing on the other. Pick five students to hold the cards at the front of the room. Line the cards up in this order (with dot numbers like 16, 8, 4, 2, 1).

Discussion

What do you notice about the number of dots on each card? (Each card has twice as many dots as the card next to it on the right.)

If we added another card to the left, how many dots would it have? (32) What about the next one after that?

We can use these cards to make different numbers. We turn some cards face down, then add up the dots on the cards that are still showing. Try asking the students to make the number 6 (using the 4-dot and 2-dot cards), then 15 (using the 8-, 4-, 2-, and 1-dot cards), then 21 (using 16, 4, and 1).

Now try counting up from zero.

Have the rest of the class watch closely to see if they notice a pattern in how the cards flip over. (Hint: each card flips half as often as the card to its right.) You might want to try this with more than one group of students.

When a card is turned face down (not showing), it stands for a zero. When it's showing, it stands for a one. This is called the binary number system—a way of writing numbers using only zeros and ones.

Ask students to arrange the cards to show 01001. What number is this in our normal counting system? (It's 9.) What would 17 look like in binary? (10001)

Try a few more examples until everyone understands.

There are five extra activities included for more practice. Students should try as many as they can.

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Worksheet Activity: Binary Numbers

Learning How to Count

So, you thought you knew how to count? Here's a brand new way to do it!

Did you know that computers only use zero and one? Everything you see or hear on a computer—words, pictures, numbers, videos, and even sounds—is stored using just those two numbers! This activity will teach you how to send secret messages to your friends using the very same method computers use.

Instructions

Cut out the cards on your sheet. Lay them out with the 16-dot card on the left, just like the example shown.

Make sure your cards stay in this same order the whole time.

Now flip the cards so that exactly 5 dots are showing—but keep the cards in the same order!

Next, figure out how to make 3, 12, and 19 using the cards. Is there more than one way to make any of these numbers?

What is the biggest number you can make? What is the smallest? Is there any number between the smallest and biggest that you can't make?

Extra for Experts: Try making the numbers 1, 2, 3, and 4 in order, one after another. Can you figure out a reliable way of flipping the cards to add one to any number?

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Photocopy Master: Binary Numbers

(Cards to cut out: 16 dots, 8 dots, 4 dots, 2 dots, 1 dot)

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Worksheet Activity: Working With Binary

The binary system uses zero and one to show whether a card is face up or not. A 0 means the card is turned over (hidden). A 1 means you can see the dots.

Can you figure out what 10101 stands for? What about 11111?

What day of the month were you born? Write that number in binary. Then find out your friends' birthdays and write those in binary too.

Try to figure out these coded numbers.

Extra for Experts: Using a set of rods that are 1, 2, 4, 8, and 16 units long, show how you can make any length from 1 to 31 units. Or, surprise an adult by showing them how a balance scale and just a few weights can weigh heavy things like suitcases or boxes!

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Worksheet Activity: Sending Secret Messages

Tom is trapped on the top floor of a department store. It's almost Christmas, and he wants to get home with his presents. He has tried calling out and even yelling, but no one is around. Across the street, he notices someone still working late at a computer. How can he get her attention?

Tom looks around for something to use. Then he has a brilliant idea—he can use the Christmas tree lights to send her a message! He finds all the lights and plugs them in so he can turn them on and off. He sends a message using a simple binary code, which he's sure the woman across the street will understand.

Can you figure out what his message says? (Use the code where each letter of the alphabet is numbered 1 through 26, and write each number in binary.)

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Worksheet Activity: E-mail and Modems

Computers connected to the internet through a device called a modem also use the binary system to send messages—but they use beeps instead of lights! A high-pitched beep stands for a one, and a low-pitched beep stands for a zero. These beeps happen so fast that all we hear is a screeching sound. If you've never heard it, try listening to a modem connecting to the internet, or call a fax machine (fax machines use modems too, to send information).

Using the same code Tom used with the Christmas lights, try sending a message to a friend. You don't need to be as fast as a real modem, though—take your time!

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Worksheet Activity: Counting Higher Than 31

Look at the binary cards again. If you added one more card to the sequence, how many dots would it have? What about the card after that? What rule are you following to make each new card?

As you can see, you only need a few cards to count very high numbers.

If you look closely at the sequence—1, 2, 4, 8, 16—you'll notice something interesting.

Try adding: 1 + 2 + 4 = ? What do you get?

Now try: 1 + 2 + 4 + 8 = ?

What happens when you add up all the numbers from the start of the sequence?

Have you heard the phrase "let your fingers do the walking"? Well, now you can let your fingers do the counting—and count much higher than ten! If you let each finger on one hand stand for one of the dot cards, you can count from 0 to 31. That's 32 different numbers (remember, zero counts too!).

Try counting in order using your fingers. A raised finger means one; a lowered finger means zero.

If you use both hands, you can count all the way from 0 to 1023—that's 1024 numbers!

If you had super bendy toes (now that would make you an alien!), you could count even higher. One hand can count to 32 numbers. Two hands can count to 32 × 32 = 1024 numbers. So what's the biggest number someone with bendy fingers and toes could reach?

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Worksheet Activity: More on Binary Numbers

1. Here's an interesting fact about binary numbers. In our normal counting system (base 10), adding a zero to the right side of a number multiplies it by 10. For example, 9 becomes 90, and 30 becomes 300.

But what happens when you add a 0 to the right side of a binary number? Try this:
1001 → 10010
(This is 9 in binary. What number does 10010 equal?)

Make up a few more examples to test your idea. What's the rule? Why do you think this happens?

2. Each card we've used so far stands for one bit on a computer ("bit" is short for "binary digit"). Our alphabet code used just five bits (five cards). But a computer also needs to tell apart capital letters, numbers, punctuation marks, and special symbols like $ or ~.

Look at a keyboard and count how many different characters a computer needs to represent. How many bits would a computer need to store all of them?

Most computers today use a system called ASCII (American Standard Code for Information Interchange), which uses a set number of bits for each character. Some countries whose languages use different alphabets need longer codes.

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What's It All About?

Computers use the binary system to store information. It's called "binary" because it only uses two different digits—also known as base two (people usually count in base 10). Each zero or one is called a bit (short for binary digit). Inside a computer, a bit is usually stored using a tiny switch called a transistor (which can be on or off) or a capacitor (which can be charged or uncharged).

When information travels over a telephone line or radio signal, high and low-pitched tones stand for the ones and zeros. On magnetic disks (like floppy disks and hard disks) and tapes, bits are stored using the direction of a magnetic field—pointing one way or the other.

Audio CDs, CD-ROMs, and DVDs store bits using light. Each tiny spot on the disc either reflects light or it doesn't.

A single bit can't hold much information by itself, so bits are usually grouped together in sets of eight. A group of eight bits is called a byte, and it can represent any number from 0 to 255.

How fast a computer works depends on how many bits it can handle at once. For example, a 32-bit computer can process 32-bit numbers in a single step. A 16-bit computer has to break those same numbers into smaller pieces first, which makes it slower.

In the end, bits and bytes are all a computer really uses to store and send numbers, words, and every other kind of information. Later activities will show how other kinds of information can also be represented this way.