DSE ICT | Data Representation and Number Systems Lesson-Prep Guide (for tutors)

DSE ICT | Data Representation and Number Systems Lesson-Prep Guide (for tutors) | Binary, hex, two’s complement, encoding

2026-11-05

Tutor lesson-prep guides · Secondary ICT

This lesson fixes one pattern: students can recite 8421, then freeze at a negative number, at ‘using 8 bits’, or at a sampling rate, and lose exactly one bit or one division by eight.

Objectives — By the end the student can (1) convert both ways with the weight method and repeated division and check the answer, (2) write a negative number in two’s complement at a stated bit width and give the representable range, and (3) compute a file size from sample rate, bit depth and channels and convert it to MB.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Powers of two	State 2^0 to 2^7	Answers must come in under five seconds: 1, 2, 4, 8, 16, 32, 64, 128
Division with remainder	Give the quotient and remainder of 45 divided by 2	A wrong remainder fails the whole question; drill three short divisions first
Bits and bytes	Say how many bits are in a byte and how many bytes in a KiB	If 1000 and 1024 are mixed up, settle KiB versus KB first

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
5 min	Opening check	Ask for 13 in 4-bit binary, then ask what happens if only 3 bits are allowed	Work alone, no notes
12 min	Weight method and repeated division	Do the same question both ways and ask why the answers agree	Copy along and say each weight aloud
10 min	Two’s complement	Start from all ones equals minus one, show invert-and-add-one, then verify the range	Write minus 42 and verify it back to 214
10 min	Encoding and file size	Cover ASCII, Unicode and UTF-8 byte counts, then compute a 60-second stereo clip	Use the formula, write units before digits
8 min	Mistakes and exit check	Fire three quick questions and redo any wrong one immediately	Answer aloud and name the step they missed

03 Teaching script (read it out)

Scene 1: Weight method: ask ‘can I subtract this?’ from the left

Write 128 64 32 16 8 4 2 1 across the page. Start from 200: 200 can take 128, write 1, remainder 72. 72 can take 64, write 1, remainder 8. 32 and 16 are too big, write 0 0. 8 fits, write 1, remainder 0. The last three bits are 0. The answer is 1100 1000. Make the student say the remainder aloud after every 1, because the marker is checking column alignment, not mental arithmetic speed.

Then have the student redo the same value by repeated division, reading remainders bottom-up. The two answers must match; if they do not, a column was dropped.

Scene 2: Two’s complement: begin with ‘all ones is minus one’

Before any rule, say this: eight bits give only 256 slots, and we lend the top 128 to negatives. So 1111 1111 is minus one and 1000 0000 is minus 128, giving a range of minus 128 to 127. For minus 42: write plus 42 as 0010 1010, invert to 1101 0101, add one to get 1101 0110. Verify by reading it unsigned, which is 214, and 214 minus 256 is minus 42. The student must run that verification once, because exam questions often follow up with the range.

Follow up: what if minus 42 must be written in 4 bits? Impossible, since 4 bits only reach minus 8, and this follow-up is what separates a level 4 script from a level 5 one.

Scene 3: Encoding and file size: units before digits

Teach encoding in three layers: ASCII uses 7 bits for 128 characters but is stored in one byte; Unicode is a code point table far larger than 256; UTF-8 is variable length, one byte for English letters and usually three for Chinese characters. For sound the formula is sample rate times bit depth times channels times seconds, divided by eight. Demonstrate 44 100 times 16 times 2 times 60 divided by 8, which is 10 584 000 bytes, then divide by 1 048 576 for about 10.1 MiB. Students forget the divide by eight because bits are not bytes.

Have the student compute 30 seconds, mono, 8-bit at 8 000 Hz. The answer is 240 000 bytes, and they must write the unit.

04 Question types and worked steps

Type	Example	Steps	Answer
Decimal to binary	Write 173 in 8-bit binary	List weights, subtract from the largest, pad to eight bits	1010 1101
Binary to decimal	Unsigned value of 1101 0110	Multiply each bit by its weight and add	128+64+16+4+2 = 214
Two’s complement	Write minus 42 in 8 bits	Write the positive, invert, add one, verify with 256	1101 0110
File size	44.1 kHz, 16-bit, stereo, 60 seconds	rate x depth x channels x seconds / 8, then divide by 1 048 576	about 10.1 MiB

Sanity check (teach this): For any width n, unsigned range is 0 to 2^n minus 1 and two’s complement range is minus 2^(n-1) to 2^(n-1) minus 1. If a binary answer needs more digits than the stated width, the question or the working is wrong, so stop and reread. After any file-size calculation, compare against the anchor that one minute of CD-quality audio is about 10 MB; a factor-of-ten gap usually means a missing divide by eight or a dropped channel.

05 Independent practice (3 questions)

Question	Answer	What to watch
Write 200 in 8-bit binary and check it by repeated division	1100 1000	Padding to eight bits and reading remainders bottom-up
Write minus 100 in 8-bit two’s complement	1001 1100	Whether invert-and-add-one was applied and verified with 256
8 000 Hz, 8-bit, mono, 30 seconds: how many bytes	240 000 bytes	Missing divide by eight, or a missing unit

06 Common mistakes and what to say

Mistake	Student writes	Say this
Too few bits	Writes 1101 for 13 and stops, though 8 bits were required	The question says 8 bits, so fill eight columns; leading zeros do not change the value.
Invert but forget to add one	Writes minus 42 as 1101 0101	You are one short after inverting. Add 42 back and see whether you land on 255.
Bits treated as bytes	Reports 10 584 000 bits as 10.5 MB	Your unit is bits. The question wants MB, so divide by eight and then by 1 048 576.
ASCII confused with Unicode	Says ASCII can store Chinese characters	ASCII only has 128 slots. Chinese needs Unicode code points, usually three bytes in UTF-8.

07 Homework (paper, section and question numbers)

Binary and hexadecimal practice (extension of section 05 here) — Do questions 1 to 3 in section 05, focusing on address segmentation
ICT chapter index — Pick the multimedia compression chapter and preview sampling versus compression
Two’s complement self-test (same type as question 2 in section 05) — Do question 2 in section 05 and show the working

08 Exit check (last 5 minutes)

Question	Answer	Decision
Largest positive and smallest negative in 8-bit two’s complement	127 and minus 128	Both must be right to pass
Roughly how many MB is 60 seconds of 44.1 kHz 16-bit stereo	About 10 MB	Answering 20 MB means a dropped channel or a missing divide by eight
Why does the same text change length in UTF-8	English takes one byte, Chinese takes several	Saying ‘variable length’ is enough

09 Tutor FAQ

Students say a calculator or program can do the conversion

The exam does not allow it, and the written working is itself a mark. Markers look for aligned weights and a verification step, not a final digit.

Should floating point be taught here

The syllabus focuses on integers and characters. Give one sentence on sign, exponent and mantissa, and spend no more than three minutes.

How do I tell real understanding from memorised answers

Ask them to reverse the question: given 1101 0110, what two values can it represent? Answering 214 and minus 42 shows understanding.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.