Euclid rectangle cuts
Record quotient and remainder until zero and verify the last positive remainder.
Adult planning suggestion: about9–12 · Logic & rule checking
- Players
- 2–3 per group
- Time
- 15–25 minutes
- Materials
- 18×12 unit rectangle on grid paper, pencil and ruler; no cutting required.
Prepare
Draw an 18-by-12 rectangle. This is a paper simulation: use grid lines and draw cuts rather than scissors. Name the larger side a and smaller b. Predict how many b-by-b squares fit across a without exceeding the rectangle.
Play
Write a=q×b+r with 0≤r<b. Shade the q squares and carry the remaining b-by-r rectangle to the next row. Replace the pair with b,r. When r is zero, stop; the last positive smaller side is the greatest common divisor.
Worked example
18=1×12+6, then 12=2×6+0, giving a final side 6. Six 6×6 squares tile the original 18×12 rectangle. Starting with 20 and 12 instead gives remainders 8,4,0 and a final side 4.
Finish and check
Check the answer divides both original lengths and that each recorded remainder is smaller than the divisor. Repeated subtraction is allowed to find q; writing an arbitrary smaller number is not. Finish after both supplied rectangles are checked.
Adaptation and limits
Work in ten trios for thirty learners: divider, recorder, checker. Use a multiplication table or adult calculation support. Advanced notation in the source is background only; our physical rectangle explanation and age choice do not imply an intervention result.
Purpose, observation and provenance
Within this activity: Record quotient and remainder until zero and verify the last positive remainder.
Ask why the chosen square side divides both original dimensions; avoid praising speed alone.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
CP Algorithms — Euclidean algorithm — Technical quotient/remainder and gcd background; our paper geometry task is original.
Age bands are planning suggestions. 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.
Player materials · Euclid rectangle cuts
Pairs: (18,12), (20,12). a ___ = q ___ × b ___ + r ___ .
Check: original side 1 / answer ___ ; side 2 / answer ___ ; square count ___ .
| a | q | b | r |
|---|---|---|---|
| 18 | ___ | 12 | ___ |
| 12 | ___ | ___ | ___ |
| 20 | ___ | 12 | ___ |
| 12 | ___ | ___ | ___ |
| ___ | ___ | ___ | ___ |
Player record · keep the host key separate
| Larger a | Divisor b | Quotient | Remainder |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host key · separate from players
(18,12) gcd 6, six squares. (20,12) gcd 4, fifteen squares.
Run the round
Draw an 18-by-12 rectangle. This is a paper simulation: use grid lines and draw cuts rather than scissors. Name the larger side a and smaller b. Predict how many b-by-b squares fit across a without exceeding the rectangle.
Write a=q×b+r with 0≤r<b. Shade the q squares and carry the remaining b-by-r rectangle to the next row. Replace the pair with b,r. When r is zero, stop; the last positive smaller side is the greatest common divisor.
18=1×12+6, then 12=2×6+0, giving a final side 6. Six 6×6 squares tile the original 18×12 rectangle. Starting with 20 and 12 instead gives remainders 8,4,0 and a final side 4.
Check the answer divides both original lengths and that each recorded remainder is smaller than the divisor. Repeated subtraction is allowed to find q; writing an arbitrary smaller number is not. Finish after both supplied rectangles are checked.
Work in ten trios for thirty learners: divider, recorder, checker. Use a multiplication table or adult calculation support. Advanced notation in the source is background only; our physical rectangle explanation and age choice do not imply an intervention result.
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