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1CP2-P-1.1 Binary patterns, bits and states
Part 1 of 5 · 1CP2-P-1 · Binary foundations
The opening Principles worksheet, building the idea of binary representation from the beginning, before any conversion or arithmetic.
Students will:
- define binary and bit, and say why two states are reliable to build with
- explain why a pattern means nothing without a context
- calculate how many patterns a given number of bits can make
- distinguish the number of states from the largest unsigned value
- repair the common error of confusing the two
Inside: 9 explanation cells, 2 multiple-choice questions, 1 number cell, 2 fill-in-the-blanks cells and 2 written answers. 18 marks, about 45 minutes.
Series: 1CP2-P-1 · Binary foundations, part 1 of 5.
Shared by Coding PathwayVerified teacher
- 16 cells
- About 45 minutes
- CC BY-SA 4.0
- Shared 17 Aug 2026
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The whole resource, exactly as a class sees it. Answers and marking are held back.
Binary patterns, bits and states
Computers represent many different things using patterns made from only two symbols. This worksheet builds that idea from the beginning, before conversion or binary arithmetic.
You will learn to define binary and bit, explain why patterns need context, calculate the number of available states and distinguish that count from the largest unsigned value.
1. Two reliable states
A digital computer processes stored instructions and data. Electronic components can reliably distinguish between two states, such as off and on. We label those states 0 and 1.
The binary system uses the two symbols 0 and 1. One bit, or binary digit, stores one of those two symbols. Computers use groups of bits to represent, process, store and transmit numbers, text, sound, graphics and program instructions.
- binary
- bit
- byte
- denary
- instructions
- states
2. A pattern needs a context
The pattern does not carry a label saying what it means. A system interprets it using an agreed encoding or representation. The same bits may therefore mean a number in one context and a character in another.
Do not try to convert the example yet. The important idea is that meaning depends on context.
Why can the same bit pattern represent different things?
- AEach bit changes its value whenever it is read.
- BThe pattern is interpreted using the rules of its current context.
- CBinary patterns can represent numbers but not other data.
- DA pattern always means the largest number it can store.
3. Count the possible patterns
List every pattern for small widths:
| Bits | Patterns | Number of patterns |
|---|---|---|
| 1 | 0, 1 | 2 |
| 2 | 00, 01, 10, 11 | 4 |
| 3 | 000, 001, 010, 011, 100, 101, 110, 111 | 8 |
Each extra bit has two choices. It attaches a 0 or a 1 to every earlier pattern, so the number of patterns doubles.
The capacity rule
For n bits, the number of unique states is 2^n.
Example for 5 bits:
2^5 = 2 × 2 × 2 × 2 × 2 = 32 states
The all-zero pattern is one valid state. Never remove it from the count.
Calculate the number of unique states for each bit width.
Use 2^n or repeated doubling. These questions ask for the count, not the maximum value.
a)How many different values can 2 bits represent?
b)How many different values can 4 bits represent?
c)How many different values can 6 bits represent?
d)How many different values can 8 bits represent?
4. States are not the same as the largest unsigned value
Unsigned values start at zero. Therefore:
| Width | Unique states, 2^n | Unsigned values | Largest unsigned value, 2^n − 1 |
|---|---|---|---|
| 3 bits | 8 | 0 to 7 | 7 |
| 4 bits | 16 | 0 to 15 | 15 |
| 8 bits | 256 | 0 to 255 | 255 |
For 8 bits, 256 answers how many patterns. 255 answers the largest unsigned value. Read the demand before calculating.
- 8
- 15
- 16
- 127
- 255
- 256
5. Explain the relationship
A complete explanation links cause and result:
Model: Increasing a pattern from 3 bits to 4 bits adds one new position. That position can be 0 or 1 for every existing 3-bit pattern, so the number of states doubles from 8 to 16.
Saying only “there are more combinations” does not explain why the total doubles.
Explain why increasing a binary pattern from 5 bits to 6 bits doubles the number of unique states.
Link the two possible values of the new bit to the old set of patterns and state the change in capacity.
Students type their answer here.
6. Repair a common error
A student writes: 8 bits give 255 patterns.
They have confused two related answers. The repair is:
- pattern count:
2^8 = 256; - values begin at 0, so the largest unsigned value is
256 − 1 = 255.
The all-zero pattern counts as one of the 256 patterns.
Which pairing is correct for an 8-bit pattern interpreted as an unsigned integer?
- A255 unique states; largest value 256
- B256 unique states; largest value 255
- C8 unique states; largest value 7
- D256 unique states; largest value 256
A door-entry system assigns each card a 6-bit unsigned identifier. Explain what gives the pattern its meaning, calculate how many different identifiers are possible and state the largest identifier value.
Give one precise point for context, one calculation or result for capacity, and one maximum value.
Students type their answer here.
Ready for unsigned conversion
You can now explain binary and bits, recognise context-dependent meaning, calculate 2^n states and distinguish state count from maximum unsigned value.
Next, the 8-bit place-value headings will show how an unsigned pattern receives a denary value between 0 and 255.