Two-state mass transition
Update both state masses simultaneously with a fixed transition rule.
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
Use two boxes A,B with initial probability mass 1 and 0. Each step sends half of A to A and half to B; all of B goes to A. These are abstract states, not a forecast of pupils, weather or illness.
Follow the rule
Calculate outgoing amounts from the old A and old B before changing either box. New A=old A/2+old B; new B=old A/2. A fraction stays a fraction; do not round to a whole person.
Check each step
The mover lays fraction slips, recorder writes both old and new values, and checker verifies nonnegative entries totaling 1. If someone uses updated A to compute B, restore the old row and redo the simultaneous step.
Worked example
From(1,0), successive distributions are(1/2,1/2),(3/4,1/4),(5/8,3/8). A mass diagram shows expected proportions, not an observed random trial with exactly those counts.
Finish and adapt
Finish after three steps; then start fresh at(0,1), obtaining(1,0),(1/2,1/2),(3/4,1/4). Discuss how the current distribution supplies the next input. This small model is not proof that a real process follows these assumptions.
Purpose, observation and provenance
Within this activity: Update both state masses simultaneously with a fixed transition rule.
Observe whether both outputs use the same old row and total mass remains 1.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Illinois CS357: Markov chains — 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.
Two-state mass transition · Player sheet
Before you begin
- Use two boxes A,B with initial probability mass 1 and 0. Each step sends half of A to A and half to B; all of B goes to A. These are abstract states, not a forecast of pupils, weather or illness.
- Calculate outgoing amounts from the old A and old B before changing either box. New A=old A/2+old B; new B=old A/2. A fraction stays a fraction; do not round to a whole person.
- The mover lays fraction slips, recorder writes both old and new values, and checker verifies nonnegative entries totaling 1. If someone uses updated A to compute B, restore the old row and redo the simultaneous step.
| Old state | To A | To B |
|---|---|---|
| A | 1/2 | 1/2 |
| B | 1 | 0 |
| Step | A | B | Total |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
| 1 | |||
| 2 | |||
| 3 |
Two-state mass transition · Host key — keep separate
(1,0)→(1/2,1/2)→(3/4,1/4)→(5/8,3/8); every row sums 1.
What to observe
Observe whether both outputs use the same old row and total mass remains 1.
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 |
|---|---|
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |