Reasoning · Grade 3-2 Weighing with Balances
Problem
Far-pan weights add; near-pan weights subtract.
This is just the idea that two sides of a balance are equal: object + (left weights) = (right weights).
3.MD.A.2Solve An Easier Related ProblemEach weight branches three ways: add, subtract, or skip.
Adding and subtracting these small numbers is grade-3 arithmetic; being systematic stops us from missing or repeating a case.
3.NBT.A.2Make A Systematic ListUsing only the 3 g weights gives 3 and 6.
Start small with one kind of weight, then build up.
3.NBT.A.2Make A Systematic ListUsing only the 8 g weights gives 8 and 16.
Same idea, now with the heavier weights.
3.NBT.A.2Make A Systematic ListMixing and subtracting adds 2, 5, 10, 11, 13, 14, 19, 22.
Putting a lighter weight on the object's side lets us reach weights BETWEEN the obvious sums, like 2 and 5.
3.OA.D.8Make A Systematic ListPutting a weight on the object's own side subtracts it, which reaches amounts that lie between the obvious sums.
A weighing says the two pans are equal, so a weight moved to the object's side counts against the object rather than for it.
Each pan's total is its weights added up, so shifting one weight across changes the reachable total by twice that weight.
Dropping zero, negatives and repeats leaves 12.
Sorting the list in order makes it easy to see there are no duplicates and to count exactly.
3.NBT.A.2Organize Information In More WaysPutting a weight on the SAME side as the object 'takes away' weight, so two 3 g and two 8 g weights can measure 12 different masses - way more than just adding them up!