Reasoning · Grade 2-1 Measurable Lengths
Problem
Forward adds and folding subtracts, so the end lands at 1 ± 3 ± 4 ± 11.
Folding a stick back is just taking that much length away, which is plain subtraction; extending it adds the length.
2.MD.A.4Draw A DiagramFolding a bar back subtracts its length and leaving it out adds it, so every arrangement is one plus-and-minus sum.
The finished chain reaches from the start to wherever the last bar ends, and that reach is built out of the individual bars laid one after another.
Folding a bar back retraces the same length in the opposite direction, and going back undoes going forward exactly.
With both 4 and 11 forward we get 19 and 13.
Adding these small numbers is ordinary within-1000 addition, just done in a tidy order so nothing is skipped.
2.NBT.B.7Make A Systematic ListWith 11 forward but 4 folded we get 11 and 5.
Working column by column with one bar flipped at a time keeps the list complete and easy to check.
2.NBT.B.7Make A Systematic ListFolding the 11 gives -3, -9, -11, -17, which as distances are 3, 9, 11, 17.
Distance is never negative, so we just read how far the end is from the start - the size of the number, ignoring the side.
2.MD.A.4Change Focus Count The ComplementCounting the repeated 11 once leaves seven: 3, 5, 9, 11, 13, 17, 19.
Sorting the results and crossing out repeats is the same as making a clean, no-repeats list.
2.MD.B.5Make A Systematic ListFolding a stick back just subtracts its length - so with a few neat plus-or-minus tries you can list every length the ruler can make, using only Grade 2 adding and subtracting!