Activity 1: Count the Dots — Binary Numbers

Summary

Computers store and send information using only two symbols: zero and one. How can words and numbers be written using just these two digits? This activity shows you how.

Curriculum Links

• Mathematics — Number: Exploring numbers written in other "bases" (counting systems). Learning to represent numbers in base two, the system computers use.
• Mathematics — Algebra: Continuing a pattern and describing the rule behind it. Looking at the patterns found in powers of two (numbers like 2, 4, 8, 16, and so on).

Skills

• Counting
• Matching
• Sequencing

Ages

7 and up

Materials

You will need a set of five binary cards (see the photocopy master) for a class demonstration. Large cards with smiley-face dot stickers work well.

Each student will need:

• A set of five cards, copied from the photocopy master onto card stock and cut out.
• The worksheet "Binary Numbers."

There are also optional extension worksheets for students who want an extra challenge:

• "Working with Binary"
• "Sending Secret Messages"
• "Fax Machines and Modems"
• "Counting Higher Than 31"
• "More on Binary Numbers"

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Binary Numbers

Introduction

Before handing out the worksheet, it helps to demonstrate the idea to the whole class first.

For this activity, you will need a set of five cards with dots on one side and nothing on the other. Choose five students to hold the cards at the front of the room, arranged in order from most dots to fewest.

Discussion

Ask the class: What do you notice about the number of dots on each card? (Each card has exactly twice as many dots as the card to its right.)

How many dots would the next card have, if we kept adding cards to the left? (32.) What about the one after that?

Now use the cards to build numbers. Turn some cards face down and add up the dots still showing. Ask 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 the 16-, 4-, and 1-dot cards).

Next, try counting upward from zero, one number at a time. Have the rest of the class watch closely to spot the pattern in how the cards flip — each card flips over half as often as the card to its right. You may want to repeat this with more than one group of students.

When a card is turned face down (hidden), it stands for a zero. When it's showing, it stands for a one. This is exactly how the binary number system works — a way of writing numbers using only 0s and 1s.

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

Practice with a few more examples until everyone understands the idea.

Five optional extension worksheets follow, letting students practice and go deeper. Students should try as many as they can.

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Worksheet Activity: Binary Numbers

Learning How to Count

Think you already know how to count? Here's a brand-new way to do it!

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

Instructions

Cut out the cards from your sheet and lay them out in order, with the 16-dot card on the far left.

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

Now flip the cards so that exactly 5 dots are showing, keeping the cards in the same order.

Try to figure out how to show 3, 12, and 19. Is there more than one way to make any of these numbers?

What is the biggest number you can make with your five cards? 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, logical method for flipping the cards to increase any number by exactly one?

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Photocopy Master: Binary Numbers

(Cards showing 16, 8, 4, 2, and 1 dots, to be cut out and used in the activities.)

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Worksheet Activity: Working With Binary

The binary system uses 0 and 1 to show whether a card is turned over or not. A 0 means the card is hidden; a 1 means you can see its dots.

Can you figure out what 10101 equals? What about 11111?

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

Try decoding a few more binary numbers on your own.

Extra for Experts: Using a set of rods with lengths 1, 2, 4, 8, and 16 units, figure out how to measure any length from 1 up to 31 units. Or try this: surprise an adult by showing them how a balance scale with just a few of these weights can weigh something heavy, like a suitcase or a box!

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Worksheet Activity: Sending Secret Messages

Tom is trapped on the top floor of a department store. It's just before Christmas, and he wants to get home to be with his family and his presents. He has tried calling out and even yelling, but no one is around to hear him. Across the street, he notices a person 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 — the Christmas tree lights! He finds them, plugs them in, and realizes he can turn them on and off to send a message. He decides to use a simple binary code that he's sure the woman across the street will understand.

Can you figure out what his message says? (Use the alphabet, where A = 1, B = 2, C = 3, and so on up to Z = 26, written 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 instead of lights, they use beeps. A high-pitched beep stands for a 1, and a low-pitched beep stands for a 0. These beeps happen so fast that all we hear is one continuous screeching sound. If you've never heard it, listen to a modem connecting to the internet, or call a fax machine (fax machines use modems too, to send information).

Using the same number code Tom used with the Christmas lights, try sending an e-mail message to a friend. You don't need to go modem-speed — take your time!

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Worksheet Activity: Counting Higher Than 31

Look again at the binary cards. 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 figure out each new card?

As you can see, just a few cards let you count to very large numbers.

Look closely at the sequence: 1, 2, 4, 8, 16...

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

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

What happens when you add up all the numbers in the sequence, starting from the beginning?

Here's a fun trick: you can count using your fingers, but go much higher than ten — no alien hands required! If you let each finger on one hand represent one binary card, 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 1, a lowered finger means 0.

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 you would need to be an alien), you could count even higher. One hand can represent 32 numbers, and two hands can represent 32 × 32 = 1024 numbers. So what's the biggest number someone with bendy toes could reach, using fingers and toes together?

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Worksheet Activity: More on Binary Numbers

1. Here's an interesting property of binary numbers. In our normal base-ten system, 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 example:

1001 (which equals 9) → 10010 (which equals... what?)

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

2. Each of the cards you've used represents one bit (short for "binary digit") in a computer. So far, our alphabet code has used just five bits. But a computer also needs to tell capital letters from lowercase ones, recognize digits and punctuation, and handle 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 is based on using this number of bits per character. Some non-English-speaking countries need longer codes to cover extra characters in their languages.

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What's It All About?

Computers today use the binary system to store and send information. It's called "binary" because it uses only two digits, and it's also known as base two (compared to base ten, which people normally use). Each 0 or 1 is called a bit (short for binary digit). Inside a computer's memory, a bit is usually stored using a transistor that is switched on or off, or a tiny component called a capacitor that is either charged or uncharged.

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

Audio CDs, CD-ROMs, and DVDs store bits optically: a spot on the disc's surface either reflects light or it doesn't, and that difference represents a 0 or a 1.

A single bit can't represent much on its own, so bits are usually grouped together in sets of eight. Eight bits together can represent any number from 0 to 255, and a group of eight bits is called a byte.

A computer's speed depends partly on how many bits it can process at once. For example, a 32-bit computer can handle a 32-bit number in a single step, while a 16-bit computer has to split that same number 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, text, and every other kind of information. Later activities will show how other kinds of information — like pictures and sound — can also be represented this way.