How many patterns?

Students count up on binary cards to find that n bits make 2ⁿ patterns and hold 0 to 2ⁿ − 1, see a count overflow, and choose how many bits a number needs (AC9TDI8K04). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

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.

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

• A student says 4 bits hold 1 to 16 (Sam's mistake). Ask: What number do the cards show when they are all face down? Fix: 0 is one of the patterns, so the 16 patterns are 0 to 15.
• A student says 10 bits make 20 patterns. Ask: What happens to the patterns when you add one card? Fix: Each card doubles them: 2, 4, 8, 16, …, so multiply.
• A student thinks overflow is an error message. Ask: What does + 1 do to 1111 if there is no fifth card? Fix: Every card flips down: the computer just loses the carry.

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.

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)

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.