Arrow resultant grid
Place arrows head to tail and compare the net displacement with the travelled route.
Adult planning suggestion: about 9–12 · Shape & spatial thinking
- Players
- 1–4 cooperatively
- Time
- 15–25 minutes
- Materials
- Square grid, arrows right 6/left 2/up 3
Prepare
Use a coordinate grid with x right and y up. Begin at(0,0). Each arrow describes a displacement in grid units; translation of an arrow must preserve its direction and length.
Play
Place right 6 from(0,0) to(6,0), left 2 from(6,0) to(4,0), and up 3 from(4,0) to(4,3). Write the endpoint of each arrow. Arrows can lie on top of a previously used segment; that does not cancel the route already travelled.
Worked example
Draw a separate resultant arrow directly from(0,0) to(4,3). Its components are(4,3). The route travelled has length 6+2+3=11 units; the resultant is not an 11-unit horizontal arrow.
Finish and check
Change the order to up 3, right 6, left 2 on a fresh grid. It ends at(4,3) too, although its intermediate points differ. A partner checks each arrow’s change in x and y rather than relying on its apparent angle.
Adaptation and limits
Finish after both routes and both resultants agree on the endpoint. As an optional geometry check the resultant length is 5 by 4²+3²=5². This paper displacement exercise measures no real force or speed and makes no experimental balance claim.
Purpose, observation and provenance
Within this activity: Place arrows head to tail and compare the net displacement with the travelled route.
Ask learners to identify the intermediate endpoints and distinguish the 11-unit route from the single resultant arrow.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
OpenStax: Graphical vector addition — Supports head-to-tail addition; our grid route is separately prepared.
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.
Rules to keep beside the materials
- Use a coordinate grid with x right and y up. Begin at(0,0). Each arrow describes a displacement in grid units; translation of an arrow must preserve its direction and length.
- Place right 6 from(0,0) to(6,0), left 2 from(6,0) to(4,0), and up 3 from(4,0) to(4,3). Write the endpoint of each arrow. Arrows can lie on top of a previously used segment; that does not cancel the route already travelled.
- Draw a separate resultant arrow directly from(0,0) to(4,3). Its components are(4,3). The route travelled has length 6+2+3=11 units; the resultant is not an 11-unit horizontal arrow.
- Change the order to up 3, right 6, left 2 on a fresh grid. It ends at(4,3) too, although its intermediate points differ. A partner checks each arrow’s change in x and y rather than relying on its apparent angle.
- Finish after both routes and both resultants agree on the endpoint. As an optional geometry check the resultant length is 5 by 4²+3²=5². This paper displacement exercise measures no real force or speed and makes no experimental balance claim.
Arrow resultant grid — play materials
For this sheet use the lower-left A1 cell corner as(0,0), grid intersections as integer coordinates; draw arrows along grid lines.
| Arrow | Δx | Δy | Endpoint |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host check — keep private until finished
Endpoint(4,3); route 11; resultant length 5 if using Euclidean length.
These are model answers or legal examples, not records of an experiment or learner outcomes.