Depth · one concept from first principles · step 2 of 4
Around the edge of a round pond, flags are set up at equal intervals of apart. What is the distance around the pond, in feet? (Ignore the thickness of the flags.)
On a closed loop the last flag joins back to the first, so gaps = flags = 31.
On a circle there is no loose end, so each flag starts exactly one gap.
3.OA.A.3Solve An Easier Related ProblemAround the closed loop of the pond, the number of gaps between neighboring flags equals the number of flags.
You can match each flag with exactly one gap — the gap that runs from that flag to the next flag as you go around the pond.
The pond edge is a closed loop, so following those gaps all the way around comes back to the first flag; the gap leaving the last flag lands on the first flag with none left over, so every flag has its own gap and every gap has its own flag.
Each of the 31 gaps is 18 feet, so the distance around is 31 times 18.
Adding up equal gaps around the edge is just multiplication, and the total is the perimeter.
3.MD.D.8Draw A DiagramOn a loop, gaps equal flags -- draw a tiny ring and Grade 3 multiplication does the rest!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →