The shape of the lesson
By the end of the lesson, learners can
- Two ladders: Open the prefixes mode of the lab and compare each decimal unit with its binary partner.
- Converting units: To move down a ladder multiply; to move up divide.
- Where each is used: Both sets of prefixes appear on real devices, which is why the numbers on the box and the numbers on the screen disagree.
- Decimal: kilo 103, mega 106, giga 109, tera 1012 (steps of 1000).
- Binary: kibi 210, mebi 220, gibi 230, tebi 240 (steps of 1024).
- Switching ladders: convert to bytes first, then divide by the other unit.
- Bits and bytes: 1 byte = 8 bits; B is bytes and b is bits.
One picture every idea returns to
Decimal prefixes go up in powers of 10 (steps of 1000): kilo = 103, mega = 106, giga = 109, tera = 1012. Binary prefixes go up in powers of 2 (steps of 1024): kibi = 210, mebi = 220, gibi = 230, tebi = 240. So 1 KiB = 1024 bytes but 1 kB = 1000 bytes. Memory sizes are naturally powers of 2, which is why the binary prefixes exist.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Kilo or kibi?. A hard drive sold as 1 TB shows up as about 931 GB on your computer, and nothing has gone missing. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Two ladders | Open the prefixes mode of the lab and compare each decimal unit with its binary partner. Project tab 6 of the lab (Binary prefixes and decimal prefixes) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Converting units | To move down a ladder multiply; to move up divide. Model the worked example Binary units on the board, one line at a time. Your turn: How many 2 MiB photos can be stored in exactly 1 GiB? | Pairs |
| 21 to 30 min | Teach and check: Where each is used | Both sets of prefixes appear on real devices, which is why the numbers on the box and the numbers on the screen disagree. | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 14 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Where each is used, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Read the symbol carefully. | MB means 10 6 bytes and MiB means 2 20 bytes. If a question uses MiB, divide by 1024 at each step, not 1000. |
| A capital B is bytes and a small b is bits. 100 Mbps is 100 000 000 bits per second, which is 12 500 000 bytes per second. | Return to Where each is used: ask a pair to state the correct version and show it on the lab before moving on. |
Binary and decimal prefixes: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
State the number of bytes in (a) one mebibyte and (b) one megabyte.
Explain the difference between a kilobyte and a kibibyte.
A computer has 16 GiB of RAM. Calculate the number of bytes. Show your working.
A file is 6 291 456 bytes. Give its size in (a) MiB and (b) MB, to two decimal places.
A customer buys a hard disk labelled 1 TB. Their operating system reports the capacity as 931 GB. Explain why, and show the calculation.
A video camera records at 25 MiB per minute onto a card with 32 GiB of free space. Calculate the number of whole minutes of video that can be recorded.
Answer key and marking notes
- (a) 220 = 1 048 576 bytes.
- (b) 106 = 1 000 000 bytes.
- A kilobyte uses a decimal prefix: 103 = 1000 bytes.
- A kibibyte uses a binary prefix: 210 = 1024 bytes, so it is slightly larger.
- 1 GiB = 230 = 1 073 741 824 bytes.
- 16 × 1 073 741 824 = 17 179 869 184 bytes (= 234).
- (a) 6 291 456 ÷ 1 048 576 = 6.00 MiB.
- (b) 6 291 456 ÷ 1 000 000 = 6.29 MB.
- The manufacturer uses decimal prefixes: 1 TB = 1012 bytes.
- The operating system divides by 230 (a gibibyte) but labels the result GB.
- 1012 ÷ 1 073 741 824 = 931.3, so no storage is missing.
- 32 GiB = 32 × 1024 = 32 768 MiB.
- 32 768 ÷ 25 = 1310.72.
- 1310 whole minutes (about 21 hours 50 minutes).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Which unit is exactly 230 bytes?
- A1 gibibyte (GiB)
- B1 gigabyte (GB)
- C1 mebibyte (MiB)
- D1 tebibyte (TiB)
How many bytes are there in 5 MiB?
- A5 242 880
- B5 000 000
- C5 120 000
- D41 943 040
A hard disk is sold as 750 GB. Approximately what capacity will an operating system that measures in GiB report?
- Aabout 698 GiB
- Babout 732 GiB
- Cabout 768 GiB
- Dexactly 750 GiB
Which quantity is most naturally an exact number of binary-prefix units?
- A16 GiB of main memory (RAM)
- BA 2 TB hard disk as labelled on the box
- CA 100 Mbps broadband connection
- DA 64 GB memory card as advertised
A 600 MiB file is downloaded over a connection that transfers exactly 100 Mbps (megabits per second). Ignoring overheads, how long does the download take?
- Aabout 50.3 seconds
- B48 seconds
- C6 seconds
- Dabout 45.8 seconds
By approximately what percentage is 1 TiB larger than 1 TB?
- Aabout 10%
- B2.4%
- Cabout 4.9%
- Dabout 7.4%
Quiz key
kibi = 210, mebi = 220, gibi = 230, so 1 GiB = 230 = 1 073 741 824 bytes. A gigabyte is the decimal unit 109 bytes, a mebibyte is only 220 bytes and a tebibyte is 240 bytes.
5 × 220 = 5 × 1 048 576 = 5 242 880 bytes. 5 000 000 is 5 MB (decimal), 5 120 000 mixes the ladders (5 × 1024 × 1000), and 41 943 040 is the number of bits, not bytes.
750 GB = 750 × 109 bytes; dividing by 230 = 1 073 741 824 gives 698.5 GiB. 732 only divides once by 1.024 (as if the units differed by a single factor of 1024/1000), 768 multiplies when it should divide, and 750 GiB ignores the difference between the two ladders.
RAM is addressed in binary, so chips hold a power of 2 bytes and 16 GiB = 16 × 230 bytes exactly. Storage manufacturers label drives and cards in decimal units (TB, GB), and network speeds such as 100 Mbps use decimal prefixes and bits.
600 × 220 × 8 = 5 033 164 800 bits; 100 Mbps = 108 bits per second; 5 033 164 800 ÷ 108 = 50.3 s. 48 s treats MiB as MB, 6 s forgets that bytes must become bits, and 45.8 s treats the megabit in the speed as 220 bits.
240 ÷ 1012 = 1.0995, so a tebibyte is about 10% larger. 2.4% is the gap between KiB and kB, 4.9% is the gap between MiB and MB, and 7.4% is the gap between GiB and GB: each step compounds another factor of 1.024.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.