Four-bit signed wrap
Decode four-bit two’s complement and separate wrapped bits from integer sums.
Adult planning suggestion: about 9–12 · Logic & rule checking
- Players
- 3 per group
- Time
- 20–30 minutes
- Materials
- Printed record sheet, pencil and paper markers
Prepare
Give each trio four bit positions weighted−8,4,2,1. Each bit is 0 or 1; their weighted sum is the signed value. The representable range is−8 through 7, not−7 through 8. Use paper cards only.
Follow the rule
To add bit strings, use ordinary binary column addition and discard carry beyond four bits. Decode the resulting four bits with the signed weights. Separately calculate the true integer sum before declaring agreement.
Check each step
Signed overflow occurs when the true sum is outside−8…7. For two positive addends, a negative decoded result signals overflow; likewise two negative inputs producing nonnegative output. Carry-out alone is not the signed-overflow test.
Worked example
0110 represents 6;0011 represents 3. Their four-bit sum 1001 decodes−7 while the true sum 9 overflows.1110 represents−2 and 0011 represents 3:0001 represents 1 with no signed overflow, despite a discarded carry.
Finish and adapt
Finish by auditing both examples and decoding 1000,1111,0111. Swap roles between decoding and adding. Do not use this representation to change real measurements or ignore overflow; the exercise explains a limited-width model.
Purpose, observation and provenance
Within this activity: Decode four-bit two’s complement and separate wrapped bits from integer sums.
Observe whether the learner distinguishes signed overflow from discarded carry.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Cornell: two’s complement — Technical background, not an evaluated primary-school lesson. This paper exercise and examples are our adaptation; age and support are adult planning suggestions.
Secondary practice: Cooperation and explaining a check.
For a class of 30, use ten groups of three. Prepare one player-material set per group and keep the host key with the adult. Rotate operator, recorder and checker after each worked example; a reader or scribe may help without choosing answers. Avoid public ranking or diagnostic labels.
Age bands are planning suggestions. Background references may address older learners and do not validate suitability for this age band. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Four-bit signed wrap · Player sheet
Before you begin
- Give each trio four bit positions weighted−8,4,2,1. Each bit is 0 or 1; their weighted sum is the signed value. The representable range is−8 through 7, not−7 through 8. Use paper cards only.
- To add bit strings, use ordinary binary column addition and discard carry beyond four bits. Decode the resulting four bits with the signed weights. Separately calculate the true integer sum before declaring agreement.
- Signed overflow occurs when the true sum is outside−8…7. For two positive addends, a negative decoded result signals overflow; likewise two negative inputs producing nonnegative output. Carry-out alone is not the signed-overflow test.
| Bits / sum | True value | Four-bit result | Overflow? |
|---|---|---|---|
| 0110 + 0011 | |||
| 1110 + 0011 | |||
| 1000 | |||
| 1111 | |||
| 0111 |
Four-bit signed wrap · Host key — keep separate
1000=−8;1111=−1;0111=7.6+3 wraps to−7 and overflows;−2+3=1 does not.
What to observe
Observe whether the learner distinguishes signed overflow from discarded carry.
Use the three-person roles described in the rules; offer reading or writing support without selecting answers.
Read the rules as well as this example. Other valid solutions may exist for the puzzle boards. This key is not a claim of uniqueness or measured learning gains.
| Constraint / action | Evidence / correction |
|---|---|
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |