Missing mass shape balances
Find one consistent unit value for each shape by checking three linked balance equations.
Adult planning suggestion: about 6–9 · Logic & rule checking
- Players
- Pairs
- Time
- 10–15 minutes
- Materials and quantities
- Per pair: one equation sheet and large paper squares, circles, triangles and unit bars. These are symbolic values, not real weights.
30 pupils: fifteen pairs; one suggests a value, the other substitutes it into all lines, then swap.
Prepare
The printed equalities are two squares plus 3 units equals 11 units; one circle plus one square equals 7; one triangle plus one circle equals 8. Every copy of a shape means the same nonnegative whole-number value. Different shapes are not required to differ unless the equations imply it.
Play
Begin with a relation you can solve. Remove equal known amounts from both sides, then share equally across identical shapes. Substitute the resulting value into another equality. Keep the original equations visible; never give one square two different values to rescue a later line.
Example
From two squares +3=11, two squares=8 and one square=4. Circle+4=7 gives circle=3. Triangle+3=8 gives triangle=5. Substitute back: 4+4+3=11, 3+4=7, 5+3=8.
Finish
Write the three shape values and check all three original lines. The group succeeds only when every line agrees. This worksheet’s “mass” is a unit model; cutting two paper shapes to the same size does not physically make them equal weight.
Adapt
Use eleven large unit marks and an adult pointing to equal groups, or let a partner write. Work through one equality before linking all three. Avoid calling a wrong trial a lack of mathematical ability.
Aim: Find one consistent unit value for each shape by checking three linked balance equations.
Practice within this activity: Keeping one value per symbol, decomposing equalities and verifying by substitution.
Adult observation: Ask the child to check an original equality rather than accept a memorised answer of four.
Mechanism and adaptation: Our version uses consistent unknown values across linked equalities. Examples and print materials are separately written; the earliest inventor is not established here.
Reference checked 24 October 2026: University of Cambridge NRICH · Getting the Balance Right — Teaching background about equality and equivalence; our three equations and answers are separate. The article does not test this worksheet.
Goals describe observable activity choices, not measured long-term gains or institutional endorsement.
Free alternatives: Draw shapes and unit marks on scrap paper. Keep paper pieces large; no small metal weights are needed.
Buying recommendations reserved; no purchase required.
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 linked equalities
Each repeated shape keeps one value. Values are units, not measured weights of paper.
○ + □ = 7
△ + ○ = 8
| □ | ○ | △ |
|---|---|---|
| ________ | ________ | ________ |
Player substitution check
| Original line | Replace all shapes | Both sides equal? |
|---|---|---|
| □+□+3=11 | ________ | ____ |
| ○+□=7 | ________ | ____ |
| △+○=8 | ________ | ____ |
Teacher instructions and checked example · separate
30 pupils: fifteen pairs; one suggests a value, the other substitutes it into all lines, then swap.
Per pair: one equation sheet and large paper squares, circles, triangles and unit bars. These are symbolic values, not real weights.
Draw shapes and unit marks on scrap paper. Keep paper pieces large; no small metal weights are needed.
Print only the relevant player pages; keep this teacher page separate. Read the full web rules before leading. A helper may write or move pieces, and anyone may pause or pass. No classroom or physical-printer trial is claimed.
Square 4, circle 3, triangle 5. Check 4+4+3=11, 3+4=7, 5+3=8. Remove 3 from both sides of the first equation, then share 8 equally between the two identical squares.