How many patterns?

Classroom resources

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Students will be able to

• I can work out how many patterns some bits can make.
• I can find the biggest number some bits can hold.
• I can explain overflow, and choose how many bits a number needs.

Timing

Predict and Try 8 min · Worked example 10 min · Vary 12 min · Practise and Make 10 min

Before class

Do the Try and Vary boards yourself first. Students need Bits and binary numbers (reading binary) and powers of 2 from maths.

The one idea: each extra bit doubles the number of patterns, so $n$ bits make $2^n$ patterns, and because one pattern is 0, the largest number is $2^n - 1$. Overflow is what happens when you count past it.

Real examples worth a minute: colour values 0 to 255 (one byte each); the micro:bit LED brightness 0 to 255; game scores and counters that wrapped round. Students who have done Arcade or micro:bit can look for 255 limits in the blocks.

Running the lesson

1. Predict: students who chose 16 counted the patterns but forgot 0; 4 counted the cards.
2. Try: ask how many presses it took to get from 0 to 1111, and how many patterns that makes.
3. Worked example: pause before line 3 and ask why we multiply, not add.
4. Vary: before Add a card, ask students to predict the new card's value and the new biggest number.
5. Make: have students say the range first (0 to 39), then the bits.

The hard part (on purpose)

Forgetting that one pattern is 0, so saying 4 bits go up to 16. Ask "what do the cards show when they are all face down?" every time a largest number comes up.

When students get stuck

Answers to the checks

1. the biggest number: You have four cards, worth 8, 4, 2 and 1. What is the **biggest** number they can show?
   15. All four cards face up: 8 + 4 + 2 + 1 = 15. There are 16 patterns, but one of them is 0, so the biggest is 15. 1111 is 15 written in binary.
2. Try: count up: Press + 1 to count up until every card is face up. What number is it?
   
   1111 = 8 + 4 + 2 + 1 = 15. Counting from 0, that took 15 presses, so with 0 there are 16 patterns.
3. Count the patterns: What is the biggest number **five** bits can show?
   31
   11111 = 16 + 8 + 4 + 2 + 1 = 31. Five bits make 2⁵ = 32 patterns: 0 to 31.
4. Find the mistake: Sam works out how many numbers 4 bits can show. Click the line where the working first goes wrong.
   line 4
   Line 4: the 16 patterns are 0000 to 1111, so the numbers are 0 to 15. Sam left out 0 (all cards face down).
5. Three bits: How many different patterns can **three** bits make?
   8
   2 × 2 × 2 = 8 patterns: 000, 001, 010, 011, 100, 101, 110, 111. That is 0 to 7.
6. One past the biggest: The board shows 1111, which is 15. Press + 1 once. What happens to the cards?
   
   Every card went face down and the count went back to 0. Four bits cannot show 16, so the count overflows.
7. Add a card: Press Add a card. What is the new card worth? Make the biggest number five cards can show.
   
   The new card is worth 16, double the 8. All five up: 16 + 8 + 4 + 2 + 1 = 31.
8. Six cards: Add one more card and make the biggest number six cards can show.
   
   The new card is worth 32. 32 + 31 = 63, which is 2⁶ − 1.
9. A byte: Eight bits are called a **byte**. What is the biggest number a byte can hold?
   255
   2⁸ = 256 patterns, so a byte holds 0 to 255.
10. How many bits?: What is the **fewest** bits that can hold the number 100?
    7
    Six bits hold 0 to 63, seven bits hold 0 to 127. 100 needs seven bits.
11. Ten bits: How many different patterns can **ten** bits make?
    1024
    $2^{10} = 1024$ patterns, so ten bits hold 0 to 1023.
12. Score overflow: An old game stores the score in one byte. The score is 255 and the player scores 1 more point. What does the score show?
    0
    255 is 11111111. Adding 1 overflows: every bit flips to 0 and the score shows 0. Real games have had bugs like this.
13. Colour values: In many programs, the amount of red in a colour goes from 0 to 255. How many bits store it?
    8
    0 to 255 is 256 values, and $2^8 = 256$. So each colour amount is one byte.
14. Three bits: What is the biggest number **three** bits can show?
    7
    111 = 4 + 2 + 1 = 7.
15. Review: read binary: Review: what number is **10101** in binary?
    21
    10101 = 16 + 4 + 1 = 21.
16. enough bits: A school gives each student in a group an ID number from 0 to 39. Use the fewest cards that can show 39, then set them to 39.
    
    Five cards stop at 31, so you need six (0 to 63). 39 = 32 + 4 + 2 + 1, so 100111.

Solutions

One possible solution for each task. Other correct programs are fine.

Vary: one change at a time › One past the biggest

Open task: any program that does what the step asks.

Vary: one change at a time › Add a card

Open task: any program that does what the step asks.

Vary: one change at a time › Six cards

Open task: any program that does what the step asks.

Vary: one change at a time › A byte

Open task: any program that does what the step asks.

Vary: one change at a time › How many bits?

Open task: any program that does what the step asks.

enough bits

Open task: any program that does what the step asks.

Stretch and support

Stretch: How many bits would you need to give every person on Earth (about 8 billion) a different number? (33: 2³² is about 4.3 billion, 2³³ about 8.6 billion.)

Support: Keep a doubling table beside the board: bits 1 to 8, patterns 2 to 256, biggest 1 to 255.

Exit ticket

How many patterns can 6 bits make, and what is the biggest number they can hold?

2⁶ = 64 patterns, so 0 to 63. The biggest is 63.

What each step is for

• the biggest number: Committing to the largest 4-bit number shows whether students forget that one pattern is 0 (16), confuse it with the number of cards (4), or read 1111 as a base-10 number.
• Try: count up: Counting up with + 1 until every card is face up settles the prediction: the count stops at 1111, which is 15.
• Worked example: count the patterns: Counting the patterns as 2 × 2 × 2 × 2 models the new move: n bits give 2ⁿ patterns and the largest is 2ⁿ − 1. Sam counts 16 patterns and then forgets 0, the misconception planted after the method.
• Vary: one change at a time: Each task changes one thing: one more past the largest (overflow), one more card (patterns double), another card, then a byte, then the edge case of choosing how many bits a number needs.
• patterns and limits: Patterns and largest numbers for other bit counts, overflow in a real context, why colour values stop at 255, and a review of reading binary from lesson 1.
• enough bits: Choosing the fewest cards for a given range reverses the process: students start from the largest number needed and work out the bits, then check it on the board.

Where it leads

Builds on: the data as numbers lessons. Colour amounts (0 to 255) are one byte each, and 16-bit audio has 2¹⁶ = 65 536 levels: ask students to explain both now.

Where it leads: data sizes (bytes, kilobytes) and, in Years 9–10, compression.

Sources

• CS Unplugged: Binary numbers (CC BY-SA 4.0) (checked 12 October 2026)