Depth · one concept from first principles · step 2 of 4
You want to join strips of tape, each long, by overlapping them by the same amount as shown in the figure, to make one long tape that is long. By how many should each overlap be?
Add up all 5 strips before any overlapping: 5 strips of 26 cm.
If they were laid end to end with no overlap, the lengths simply add.
3.OA.A.3Identify SubproblemsThe finished tape is 110 cm. Overlapping makes it shorter, so the amount lost is the separate total minus the final length.
Every overlap hides some tape; the missing 20 cm is all the hidden overlap together.
3.MD.D.8Draw A DiagramThe orange regions sit between neighboring strips. With 5 strips there are 4 such overlaps (one fewer than the number of strips).
Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.
3.OA.D.9Look For A PatternJoining 5 strips in a row makes exactly 4 overlaps, one fewer than the number of strips.
The first strip sits at the front with nothing before it, so it makes no overlap, while every strip after it laps onto the one already placed and makes exactly one overlap.
Each later strip covers part of just its single left-hand neighbor, so the overlaps match the strips-after-the-first one for one.
The 5 strips break cleanly into the one first strip and the strips after it, so the number after the first is what is left when 5 is separated into those two parts.
No strip is counted twice and none is missed, so the one first strip and the rest add back to all 5, which leaves the rest as 5 take away 1.
The 20 cm of hidden tape is shared equally among the 4 overlaps.
Equal overlaps means equal sharing, so divide the total overlap by how many there are.
3.OA.A.3Identify SubproblemsWhen you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition. Why the shapes repeat →