The shape of the lesson
By the end of the lesson, learners can
- Why binary?: Click the bits in the lab to switch them on and off.
- Binary to denary: Write the place values above the bits, then add up the place values of every 1.
- Denary to binary: Two methods give the same answer.
- Place values: 128, 64, 32, 16, 8, 4, 2, 1 for 8 bits; keep doubling for 16 bits.
- Binary to denary: add the place values of the 1s.
- Denary to binary: subtract the largest place values that fit, or divide by 2 and read the remainders upwards.
- Check: convert your answer back the other way.
One picture every idea returns to
Computers process data using logic gates and store it in registers, both made of switches that are either off or on. So all data is represented in binary (base 2), which uses only the digits 0 and 1. People use denary (base 10). In binary each place value is twice the one to its right: 1, 2, 4, 8, 16, 32, 64, 128 ... A single binary digit is a bit; 8 bits make a byte.
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, Two symbols are enough.. A computer is built from billions of tiny switches that are either off or on. 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: Why binary? | Click the bits in the lab to switch them on and off. Project tab 1 of the lab (Binary and denary converter) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Binary to denary | Write the place values above the bits, then add up the place values of every 1. Model the worked example 8 bits on the board, one line at a time. Your turn: Convert 11001001 to denary. | Pairs |
| 21 to 30 min | Teach and check: Denary to binary | Two methods give the same answer. Model the worked example Subtract the place values on the board, one line at a time. Your turn: Convert 58 to 8-bit binary. | Round Table |
| 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, 13 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 Denary to binary, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| "Computers use binary because it is easier for them" earns nothing. | Say why: the computer is made of switches (logic gates and registers) with only two states, on and off. |
| Leaving out leading zeros changes nothing in value, but if the question says "8-bit" you must give all 8 bits: 00111010, not 111010. | Return to Denary to binary: ask a pair to state the correct version and show it on the lab before moving on. |
Why computers use binary: binary and denary: worksheet
6 questions, 13 marks. Written in the style of the papers; no past paper question is reproduced.
Explain why computers represent all data in binary.
Convert the binary number 10110110 to denary. Show your working.
Convert the denary number 77 to 8-bit binary.
Convert the denary number 1000 to 16-bit binary.
A sensor stores readings as 8-bit unsigned binary integers. (a) State the smallest and largest readings it can store. (b) A reading of 300 is taken. Explain why it cannot be stored.
A student says 1101 in binary is one thousand one hundred and one. Explain the mistake and give the correct denary value.
Answer key and marking notes
- Computers process and store data using logic gates and registers made of switches (transistors).
- Each switch has only two states, off and on, which are represented by 0 and 1.
- Place values of the 1s: 128, 32, 16, 4, 2.
- 128 + 32 + 16 + 4 + 2 = 182.
- 77 = 64 + 8 + 4 + 1.
- 01001101 (8 bits, with the leading zero).
- 1000 = 512 + 256 + 128 + 64 + 32 + 8.
- 0000 0011 1110 1000.
- (a) Smallest 0 (00000000); largest 255 (11111111).
- (b) 300 is larger than 255, the biggest value 8 bits can hold.
- It would need 9 bits (300 = 100101100).
- Binary place values are 8, 4, 2, 1, not 1000, 100, 10, 1.
- 1101 = 8 + 4 + 1 = 13.
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.
Why do computers represent all data in binary?
- ATheir logic gates and registers are made of switches that have only two states, off and on
- BBinary numbers always take up less storage space than the same numbers in denary
- CBinary is easier than denary for programmers to read and write
- DDenary cannot be used to represent numbers larger than 255
Convert the 8-bit binary number 10010110 to denary.
- A150
- B105
- C151
- D86
Convert the denary number 75 to 8-bit binary.
- A
01001011 - B
01001101 - C
11010010 - D
01010011
What is the largest denary number that can be stored in an 8-bit register?
- A255
- B256
- C128
- D127
Convert the 16-bit binary number 0000 0010 0101 1000 to denary.
- A600
- B88
- C536
- D604
What is the smallest number of bits needed to store the denary number 300 as an unsigned binary integer?
- A9
- B8
- C3
- D10
Quiz key
Logic gates and registers are built from switches that are either off (0) or on (1), so data must be held as binary digits. Binary actually needs more digits than denary for the same number, so it does not save space. Humans find binary harder to read, which is why hexadecimal is used. Denary has no 255 limit; that is the limit of an 8-bit binary register.
The 1s sit under 128, 16, 4 and 2: 128 + 16 + 4 + 2 = 150. 105 comes from reading the bits in reverse order (01101001). 151 wrongly includes the 1 place value, which holds a 0. 86 misses the 128 bit and adds 64 instead (64 + 16 + 4 + 2).
75 = 64 + 8 + 2 + 1, so the bits under 64, 8, 2 and 1 are 1: 01001011. 01001101 is 77 (64 + 8 + 4 + 1). 11010010 is the correct bits written backwards (210). 01010011 is 83 (64 + 16 + 2 + 1).
11111111 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. There are 256 different values, but they run from 0 to 255, so 256 itself does not fit. 128 is only the value of the most significant bit. 127 is 01111111, which leaves the top bit unused.
The 1s are under 512, 64, 16 and 8: 512 + 64 + 16 + 8 = 600. 88 converts only the lower 8 bits and ignores the 512 bit. 536 is 512 + 16 + 8, missing the 64. 604 wrongly adds a 4 that is not set.
8 bits can hold at most 255, which is too small. 9 bits hold up to 511, so 300 (100101100) needs 9 bits. 3 is the number of denary digits, not binary digits. 10 bits would work but is not the smallest number.
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.