Reasoning · Grade 5-2 Parity
Problem
Turning one coin makes its face alternate.
This is just pairing turns off two at a time — the same idea as deciding whether a group of objects is odd or even by pairing them up. Each pair of turns cancels itself out, and what is left over (nothing, or one lone turn) decides the face.
2.OA.C.3Create A Physical RepresentationThe per-coin counts add to the total.
Re-recording the messy sequence as 3 tallies is the move that makes the problem small. A fourth grader can already see that if you sort 11 turns into 3 boxes, the box totals add back up to 11.
2.OA.A.1Organize Information In More WaysEven turns leave heads, odd turns leave tails.
Adding whole numbers follows a fixed odd/even pattern — even + even is even, odd + even is odd, odd + odd is even — so knowing the parity of the total tells you something certain about the parts even when you cannot see the parts.
3.OA.D.9Look For A PatternA coin's face depends only on how many times it has been turned, not on when those turns happened.
Each turn swaps the coin between its two faces, so two turns cancel and only the odd or even count survives.
The whole sequence of moves is its individual turns put together, so the order they came in cannot affect the end state.
In (1) the hidden coin's count is odd.
You never need to know the actual counts. Two evens can only add to an even, and an odd total minus an even part always leaves an odd part — that single fact pins down the hidden coin.
3.OA.D.9Look For A PatternSo the second coin shows tails.
Building one easy example that matches the picture is a cheap way to make sure the parity argument was not misread — and it costs only an addition within 20.
2.OA.A.1Solve An Easier Related ProblemIn (2) the other two counts add to an odd number.
The odd/even rules for adding two numbers are exactly the ones learned with counters: odd + odd and even + even both pair up perfectly, so an odd sum can only come from one odd and one even.
3.OA.D.9Look For A PatternSo those two show one of each face.
Two examples that both fit the picture but disagree about which coin is tails prove that 'different faces' is the most that can be known — a good habit is to check that the question only asked for that much.
2.OA.C.3Solve An Easier Related ProblemA coin only remembers how many times it was turned, not when — so the odd-or-even of the total number of turns tells you about the coins you cannot even see.