Counting with light bulbs
Light bulbs are only on or off. Can they show any number? What is each one worth?
Board
- Number on the board
- 0
Under each bulb is its value, or ? until you find it.
Find what each bulb is worth
Tap the bulbs to turn them on and off, and watch the number on the board. Each bulb that is on adds a hidden value. Start with the two on the right.
The bulbs on the left unlock later: first you find the two on the right and guess the value of the next one.
Then tap Count from 0 and notice which bulb switches most often.
Make 13
Turn on the right bulbs so the board shows 13, then tap Check. No Add 1 and no counting: pick the bulbs yourself.
Write 19 in 0s and 1s
Without touching the board, write 19 in base 2: one digit for each bulb, 1 for on and 0 for off.
The largest with 5 bulbs
Before you turn any on, guess: what is the largest number 5 bulbs can make?
Then turn on all 5 bulbs and check the board.
Make your guess first. The Check the board button unlocks when you tap Lock in my guess or I don't know.
One more bulb
If the 5-bulb board got a 6th bulb on the left, what would it be worth?
A byte
With 8 bulbs, what is the largest number on the board?
Bulbs with 3 states
Change States per bulb to 3. Each bulb can be off, dim, or bright, and the values are hidden again. Find the new values using dim light.
With 5 bulbs that have 3 states, what is the largest number on the board?
Counting table
| Number | Bulbs in 0s and 1s |
|---|
What the board does
The explanation unlocks when you finish task 1, finding what each bulb is worth.
Each bulb is worth twice the one to its right. You already use this idea every day, only with 10 instead of 2. In 4,572, each place is worth 10 times the place to its right: ones, tens, hundreds, thousands. The 4 is worth 4,000 because it sits in the thousands place.
| Place, from right to left | Base 10 (× 10) | Bulbs (× 2) |
|---|---|---|
| 1st | 1 | 1 |
| 2nd | 10 | 2 |
| 3rd | 100 | 4 |
| 4th | 1,000 | 8 |
| 5th | 10,000 | 16 |
The bulbs make up the binary system, or base 2: a way of writing numbers where each place is worth 2 times the place to its right, and each place has only two digits, 0 (off) and 1 (on). The place values are powers of 2: , , , , and the rightmost place is worth 1. 11 is 8 + 2 + 1, so in base 2 it is written (a 0 at the far left can be dropped, as in 011):
So yes, bulbs can show any number, as long as there are enough of them: each extra bulb doubles how many numbers the board can show. Computers store everything this way, with circuits that are on or off. Each bulb is a bit, a place that can only be 0 or 1. Eight bits together make a byte. Switch the board to 8 bulbs and see how high it goes.
For teachers
Objective
By the end, students can write a number in base 2, knowing that each place is worth twice the place to its right. The activity starts from the place value the class already knows in base 10 and shows the same rule with 2 in place of 10.
Prerequisites
Place value in the decimal system (ones, tens, hundreds, thousands), addition and multiplication of whole numbers, and powers with whole-number exponents. No computer science knowledge is needed.
How to use in class
- Project the board with the values hidden and read the opening question. Ask for guesses: what is each bulb worth? Write them on the board (2 to 3 minutes).
- Let the class work in pairs until they find all five values (task 1 in the panel, 5 minutes). The two bulbs on the right start unlocked; the 3rd and 4th unlock only after the pair locks in a guess for each one. Ask who guessed 3 for the 3rd, and why. The strategy that works is turning on a single bulb.
- Tap Count from 0 and pause at 7 and at 8. Ask what happened to the bulbs in that step and compare it with 99 turning into 100 on a car's odometer (3 minutes).
- Read the explanation with the class (it unlocks when task 1 is done), paying attention to the table that compares base 10 with the bulbs (5 minutes).
- Challenge in pairs, tasks 2 to 6 in the panel next to the board (8 minutes). In task 4, each pair locks in a guess before turning on the bulbs. For those who finish early, the last task, Going further.
- Wrap-up: why is the largest number with 5 bulbs 31, and not 32? (3 minutes)
Common mistakes
Thinking the values go up by 1 (1, 2, 3, 4, 5). It shows up as a guess of 3 for the 3rd bulb (or 5 for the 4th), a guess of 15 in task 4, and an answer of 6 in task 5.
Counting the bulbs that are on instead of adding their values. The student guesses 5 for the largest number with 5 bulbs.
Reading the board backwards, as if the leftmost bulb were worth 1. In task 2, the student makes 22 instead of 13; in task 3, they write 11001 instead of 10011. The activity flags both cases.
Mixing up the largest number with the value of the next place: 32 instead of 31, 256 instead of 255. It is the same mistake as thinking the largest 2-digit number is 100.
Discussion questions
- Why doesn't base 2 have the digit 2? And why doesn't base 10 have a digit for 10?
- With 5 bulbs the largest is 31, and with 6 it is 63. What rule connects the number of bulbs to the largest number?
- When counting, the bulb on the right switches with every number. How often does the second one switch? And the third?
- A byte holds 256 different numbers. A screen color usually uses one byte for red, one for green, and one for blue. How many colors can you make?
- If people had 8 fingers, how do you think we would write numbers?
Based on Contemporary Mathematics, OpenStax, sections 4.1 and 4.3, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.