Distributive token groups
Compare three identical bundles with their expanded variable and unit counts.
Adult planning suggestion: about 9–12 · Logic & rule checking
- Players
- 1–4 with adult support
- Time
- 15–25 minutes
- Materials
- Six x labels and twelve unit labels
Prepare
A token marked x means an unspecified number; a unit token means 1. Do not assume the printed width of an x token is the value of x. Prepare three bundles, each containing one x and two units.
Play
Write the bundled expression 3(x+2). Open all three bundles without creating or removing tokens. Count three x tokens and six units, then write 3 x+6. The outside 3 multiplies both terms inside.
Worked example
For x=4, each bundle is 6 and all three total 18. The expanded form is 3×4+6=18. The incorrect 3 x+2 would give 14 and has failed to count four of the unit tokens.
Finish and check
Build a second independent example 2(2 x+3): each of two groups has two x tokens and three units. The total is 4 x+6. Do not turn two x tokens into x squared: repeated addition is not multiplication of variables.
Adaptation and limits
Finish after a partner matches every original token to the expanded collection and checks both expressions at x=4. Our tokens model symbolic sums, not accurate measured rectangle areas; hand-written labels fully replace cut cards.
Purpose, observation and provenance
Within this activity: Compare three identical bundles with their expanded variable and unit counts.
Match every unit and x token before and after expansion; record any missing units in an incorrect expression.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
OpenStax: Distributive Property — Reference supports the stated concept; our fictional numbers, instructions and paper diagrams are independently prepared, not reproduced teaching pages.
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
- A token marked x means an unspecified number; a unit token means 1. Do not assume the printed width of an x token is the value of x. Prepare three bundles, each containing one x and two units.
- Write the bundled expression 3(x+2). Open all three bundles without creating or removing tokens. Count three x tokens and six units, then write 3 x+6. The outside 3 multiplies both terms inside.
- For x=4, each bundle is 6 and all three total 18. The expanded form is 3×4+6=18. The incorrect 3 x+2 would give 14 and has failed to count four of the unit tokens.
- Build a second independent example 2(2 x+3): each of two groups has two x tokens and three units. The total is 4 x+6. Do not turn two x tokens into x squared: repeated addition is not multiplication of variables.
- Finish after a partner matches every original token to the expanded collection and checks both expressions at x=4. Our tokens model symbolic sums, not accurate measured rectangle areas; hand-written labels fully replace cut cards.
Distributive token groups — player materials
| Expression | x tokens | Unit tokens | At x4 |
|---|---|---|---|
| 3(x+2) | |||
| 2(2x+3) | |||
| 3x+2 |
Host check — keep separate
3(x+2)=3 x+6;2(2 x+3)=4 x+6; at x 4 totals 18 and 22.
Examples are stipulated model cases, not measured experimental or learning outcomes.