Students must be able to represent decimal values between 0 and 255 in…

The idea

Eight binary columns give place values 128, 64, 32, 16, 8, 4, 2 and 1, so any decimal from 0 to 255 can be written with exactly eight bits. To convert, work left to right: if the place value fits into the remaining amount, write 1 and subtract it; otherwise write 0. For 200, 128 fits (remainder 72), 64 fits (remainder 8), 32 and 16 do not, 8 fits (remainder 0), and the last three columns are 0, giving 11001000. Check by adding the place values of the 1s back to the original number. The limits matter: 0 is 00000000 and 255 is 11111111, so nine bits would be needed for 256.

What earns the marks

• Recognises the eight place values 128, 64, 32, 16, 8, 4, 2 and 1 and that eight bits span 0 to 255.
• Converts a given decimal correctly by subtracting place values from left to right and recording 1 or 0 in each column.
• Verifies an answer by summing the place values of the 1 bits to recover the original decimal.
• Identifies boundary cases: 0 as 00000000 and 255 as 11111111, and explains why 256 needs a ninth bit.
• Keeps all eight positions, including leading zeros, so the result is a full 8-bit pattern.

Common mistakes and how to put them right

• Writing bits right to left in the wrong order.

  Put it right: Fix the place values above the columns first, largest on the left, then fill each column.

• Dropping leading zeros so 5 becomes 101 rather than 00000101.

  Put it right: Always show all eight positions for a 0 to 255 value.

• Adding place values instead of subtracting as each bit is set.

  Put it right: Track the running remainder and only set a bit when its place value fits.

Exam tips

• Write the place values 128 to 1 above your working before converting, so each bit is placed deliberately.
• Always check by adding the place values of the 1s; this catches most slips quickly.
• For multiple-choice items, eliminate options with more than eight bits or with a value-sum that does not match the target decimal.