Depth · one concept from first principles · step 2 of 4
Around the edge of a pond, trees are planted at equal intervals of m apart. What is the distance all the way around the pond, in meters?
(Figure) A top-down view of a round (oval) pond. Five trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled m.
Because the path closes back to the start, the number of gaps between trees equals the number of trees: 5 trees make 5 gaps.
On a circle every tree has a gap leading to the next, with no leftover end, so gaps match trees.
3.OA.D.9Draw A DiagramOn the closed path around the pond, the number of gaps between neighboring trees equals the number of trees.
Match each tree with the one gap that leaves it and heads to its next neighbor; each tree gets exactly one such gap and no two trees share it.
Because the trees go all the way around and the path closes back to the first tree, the gap leaving the last tree lands on the first tree instead of running off an end, so every gap gets a tree and every tree gets a gap with none left over.
Each of the 5 equal gaps is 14 m, so the distance all the way around is 5 times 14.
Equal gaps mean equal-size jumps, so the total is just 5 copies of 14 m.
3.OA.A.3Look For A PatternAround a closed loop the gaps equal the trees, so 5 gaps of 14 m means 5 times 14 = 70 m!
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The same shape shows up later on the AMC, as Pair-Sum Invariant. Why the shapes repeat →