Reasoning · Grade 6-1 Truth and Falsehood
Problem
A score counts the questions got right.
Turning a score into a count of matching columns is the same move as counting how many items in a list have a property — nothing beyond simple counting is needed to make the table say something useful.
2.OA.A.1Introduce A VariableA disagreeing column gives the pair exactly 1.
Because there are only two possible answers, disagreement is already an answer of sorts — a second grader can see that if one of us wrote O and the other wrote X, exactly one of us got the point, without knowing which mark is correct.
2.OA.A.1Change Focus Count The ComplementA column where two students disagree is worth exactly one point to the pair, whichever of them is right.
With only two possible marks, one of the two students matches the key and the other does not, and never both or neither.
Each column contributes to the pair's combined score once and only once, so the totals can be read column by column.
One comparison fixes question 1.
This is a missing-addend question, 3 + ? = 5, and the only possible values for the missing part are 0 and 2 — so the arithmetic itself chooses between both right and both wrong.
1.OA.D.8Change Focus Count The ComplementAnother comparison fixes questions 3 and 4.
Once you know a column is worth 0 to a pair who marked it the same way, you know the key disagrees with both of them — and with only two marks available, disagreeing with X leaves only O.
2.OA.A.1Introduce A VariableThe leftover score finishes question 2.
A total of 1 point spent on one question leaves nothing for the others, so every remaining mark of Ethan's has to be wrong — subtraction alone settles the last column.
1.OA.D.8Introduce A VariableOf 16 possible keys only one fits all three scores.
Sixteen cases is a short enough list to write out completely, and checking every one of them is what upgrades a good deduction into a certainty.
2.OA.A.1Make A Systematic ListWhen two answer sheets disagree on a question, exactly one of them earned that point — so adding two scores and subtracting tells you the key without guessing.