Activity 1: Count the Dots — Binary Numbers

Summary

Computers store and send all data as a series of zeros and ones. How can something with only two symbols represent words and numbers?

Curriculum Links

• Mathematics: Number — Exploring numbers in other bases. Representing numbers in base two (a number system using only two digits, 0 and 1, instead of the ten digits we normally use).
• Mathematics: Algebra — Continuing a sequential pattern and describing a rule for it. Patterns and relationships in powers of two.

Skills: Counting, matching, sequencing

Ages: 7 and up

Materials

You will need a set of five binary cards (see page 6) for the demonstration. A4-sized cards with smiley-face sticker dots work well.

Each student will need:

• A set of five cards (copy the Photocopy Master on page 6 onto card stock and cut out)
• Worksheet Activity: Binary Numbers (page 5)

Optional extension activities require:

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

---

Binary Numbers

Introduction

Before handing out the worksheet on page 5, it helps to demonstrate the idea to the whole class first.

You'll need a set of five cards, each with dots on one side and nothing on the other. Pick five students to hold the demonstration cards at the front of the room, arranged in order.

Discussion

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

If we kept adding cards to the left, how many dots would the next one have? (32) The one after that?

You can use these cards to build numbers by flipping some face-down and adding up the dots still showing. Have students make 6 (using the 4-dot and 2-dot cards), then 15 (8, 4, 2, and 1), then 21 (16, 4, and 1).

Now try counting up from zero. The rest of the class should watch closely to spot a pattern in how the cards flip — each card flips half as often as the card to its right. Try this with more than one group if time allows.

When a card is hidden, it stands for a zero. When it's showing, it stands for a one. This is the binary number system — a way of writing numbers using only 0s and 1s.

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

Practice a few more examples until the concept clicks.

Five optional extension activities follow for extra practice. Students should attempt as many as they can.

---

Worksheet Activity: Binary Numbers

Learning how to count

Thought you knew how to count? Here's a 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 to send secret messages to your friends the same way a computer stores information.

Instructions

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

Make sure your cards stay in that exact order. Now flip cards so that exactly 5 dots are showing (keeping the order the same).

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

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

Extra for Experts: Try making 1, 2, 3, and 4 in order. Can you figure out a reliable method for flipping the cards to increase any number by exactly one?

---

Photocopy Master: Binary Numbers

(Card template for cutting out — five cards showing 16, 8, 4, 2, and 1 dots)

---

Worksheet Activity: Working With Binary

The binary system uses 0 and 1 to show whether a card is face up 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 it in binary. Find out your friends' birth dates in binary too.

Try decoding these numbers.

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

---

Worksheet Activity: Sending Secret Messages

Tom is trapped on the top floor of a department store just before Christmas, trying to get home with his presents. Calling and yelling haven't worked, and no one's around — but across the street, he spots someone still working late at a computer. How can he get her attention?

Tom looks around for ideas. Then it hits him: he can use the Christmas tree lights! He plugs them all in so he can switch them on and off, then uses a simple binary code he's sure the woman across the street will understand. Can you crack it?

(A number-to-letter key is provided, assigning 1–26 to the letters a–z.)

---

Worksheet Activity: E-mail and Modems

Computers connected to the internet through a modem (a device that converts computer data into signals that can travel over phone lines) also use binary — but with beeps instead of lights. A high-pitched beep stands for a one; a low-pitched beep stands for a zero. These beeps happen so fast that all we hear is a 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.

Using the same code Tom used in the department store, try sending an email message to a friend. Go easy on yourselves, though — you don't need to be as fast as a real modem!

---

Worksheet Activity: Counting Higher Than 31

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

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

Look closely at this 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?

Ever heard the phrase "let your fingers do the walking"? Now you can let your fingers do the counting — and get much higher than ten, no alien hands required! Let each finger on one hand represent one dot-card. Using binary, you can count from 0 to 31 on one hand alone — that's 32 numbers (remember, zero counts too!).

Try counting in order with your fingers: a raised finger means one, a lowered finger means zero.

Using both hands, you can count all the way from 0 to 1023 — that's 1024 numbers! If you had extremely flexible toes (now that would take an alien), you could count even higher. Since one hand gives you 32 numbers, and two hands give you 32 × 32 = 1024 numbers, what's the biggest number "Miss Flexi-Toes" could reach using her toes as well?

---

Worksheet Activity: More on Binary Numbers

1. Here's an interesting property of binary numbers: what happens when you add a zero to the right side of a number? In our normal base-ten system, adding a zero to the right multiplies the number by 10 (9 becomes 90, 30 becomes 300).

But what happens when you add a 0 to the right of a binary number? Try this:

1001 → 10010
(9) → (?)

Test a few more examples. What's the rule? Why do you think this happens?

2. Each card we've used represents one bit on a computer (short for "binary digit"). Our alphabet code so far uses just five bits. But a computer also has to distinguish capital letters, digits, 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 be needed to store all of them?

Most computers today use a system called ASCII (American Standard Code for Information Interchange), built around this number of bits per character — though some non-English-speaking countries need longer codes to cover their characters.

---

What's It All About?

Computers use the binary system to represent information. It's called binary because it relies on only two digits — also known as base two (humans typically use base ten). Each 0 or 1 is called a bit. Inside a computer's memory, a bit is usually represented by a transistor switched on or off, or a capacitor either charged or uncharged.

When data travels over a phone line or radio signal, high and low-pitched tones stand in for ones and zeros. On magnetic disks (like hard drives) and tapes, bits are represented by the direction of a magnetic field on a coated surface — either north-to-south or south-to-north.

Audio CDs, CD-ROMs, and DVDs store bits optically: each spot on the surface either reflects light or it doesn't.

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

A computer's speed depends partly on how many bits it can process at once. A 32-bit computer, for example, can handle 32-bit numbers in a single operation, while a 16-bit computer has to break those same numbers into smaller pieces first — making it slower.

In the end, bits and bytes are the building blocks computers use to store and transmit numbers, text, and every other kind of information. Later activities will show how other types of information can also be represented this way.