Balance Puzzles: Separating 'I Can't Solve It' from 'It Can't Be Solved'
Ms. Brennan's small group is working on balance puzzles built from three shapes: a triangle, a square, and a circle. She gives four puzzles and asks each of her six students to write an answer plus one line of justification. Puzzle 1: two triangles balance one square; three squares balance two circles; one triangle, one square, and one circle together weigh 36 grams. How much does each shape weigh? Four students answered 6, 12, and 18 with clear substitution. Puzzle 2: a square plus a triangle weigh 15 grams; a square plus a circle weigh 19 grams; a triangle plus a circle weigh 16 grams. How much does a circle weigh? Two students answered 10, three left it blank, one wrote 'cannot tell.' Puzzle 3: two triangles weigh more than one square, and one square weighs 10 grams. What does one triangle weigh? Five students wrote 6 or 7; one wrote 'more than 5 grams.' Puzzle 4: a star equals three moons; a moon equals four dots. How many dots equal two stars? All six answered 24.
1. Solve Puzzles 1 and 2 completely, and identify the step that makes Puzzle 2 solvable where substitution alone stalls.
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Puzzle 1. Two triangles balance one square, so s = 2t. Three squares balance two circles, so 3s = 2c, giving c = 1.5s = 3t. Substituting into the total: t + 2t + 3t = 6t = 36, so t = 6 g, s = 12 g, c = 18 g. Check against both balance statements: 2(6) = 12 and 3(12) = 36 = 2(18).
Puzzle 2. Write the three statements as s + t = 15, s + c = 19, t + c = 16. Substituting one into another is awkward because each equation omits a different shape. Instead add all three: 2s + 2t + 2c = 50, so s + t + c = 25. A circle is then the total minus the pair that excludes it: c = 25 − (s + t) = 25 − 15 = 10 g. Check: s = 25 − 16 = 9, t = 25 − 19 = 6, and 9 + 10 = 19 as required.
The move is symmetry: when each equation is missing a different variable, sum the equations first. The three students who left it blank were likely committed to isolating one shape at a time, which has no foothold here.
2. Five students gave a number for Puzzle 3. Explain why the single student who wrote 'more than 5 grams' is the only correct response, and name the habit the other five displayed.
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The information given is 'two triangles weigh more than one square' and 'one square weighs 10 grams.' Therefore 2t > 10, so t > 5. That is the complete consequence - the constraint is an inequality, and it pins down a range, not a point. Any weight above 5 grams satisfies it, so no single number is correct.
The other five treated 'more than' as 'equals.' Having found 2t = 10 comfortable, they solved for 5 and then adjusted upward to 6 or 7 to respect the word 'more.' That is a sensible repair, but it fabricates a value the puzzle never supplies: nothing rules out 5.5, 8, or 40. This is the same reflex that makes students answer 'cannot be determined' questions with a number - the format of the sheet implies one right number, so a number gets produced.
Note that this is not a computational error and would not show up on a word-problem test with unique answers. It is a habit about what kind of answer a problem is allowed to have.
3. Which puzzle or puzzle type should Ms. Brennan use as an exit ticket to check whether students can distinguish 'insufficient information' from 'I cannot do it'? Justify using the structure of the puzzles.
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Build the exit ticket on Puzzle 3's structure, not its numbers: give an inequality plus one known quantity so that the answer is a bounded statement rather than a value. For instance: 'Two circles weigh less than three squares; one circle weighs 12 grams. What does one square weigh?' The student must produce 'more than 8 grams' and must resist writing 18 or 9.
Why this and not Puzzle 2? Puzzle 2 also stalls a naive solver, but it is solvable once the equations are summed, so a blank there confuses two different diagnoses: missing a technique, or believing the problem is unsolvable. Those call for different follow-ups. Puzzle 3 has a genuinely open answer, so a numerical response is unambiguous evidence that the student overrode the wording with an assumption.
Puzzle 1 and Puzzle 4 are the wrong instruments entirely - both have single correct numeric answers and are already at ceiling, so they confirm nothing. A useful pair for one sitting is a Puzzle 3 variant plus Puzzle 2: the first shows whether the student can tolerate an unclosed answer, the second whether they can find the technique that closes a solvable one.