Problem
Posts versus gaps on lines and loops
Thirty-one flags stand at equal 18-foot gaps around a round pond. Find the distance all the way around the pond.
OperationsMeasurement & data
Your answer
ft
Watch out Don't subtract 1 like you would for a straight line of posts -- on a closed loop, the number of gaps equals the number of flags, not one fewer.
How to solve
Strategy Draw a Diagram — Sketch a few flags around a loop to see that on a closed path the number of gaps equals the number of flags. Then the total distance is the number of gaps times 18 feet.
1STEP 1
On a loop, gaps equal flags
On a closed loop the last flag joins back to the first, so gaps = flags = 31.
number of gaps = number of 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.
Why?
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.
Why?
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.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Multiply gaps by the gap length
Each of the 31 gaps is 18 feet, so the distance around is 31 times 18.
31 × 18 = 31 × 20 - 31 × 2 = 620 - 62 = 558
Adding up equal gaps around the edge is just multiplication, and the total is the perimeter.
3.MD.D.8Draw A DiagramAnswer
558 ft
31 × 18 = 558
About 30 gaps of about 20 feet would be roughly 600 feet; our exact 558 feet is close to that estimate, so it is reasonable.
Takeaway
On a loop, gaps equal flags -- draw a tiny ring and Grade 3 multiplication does the rest!
- On a loop, gaps equal flags
- Multiply gaps by the gap length
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