Posts versus gaps on lines and loops

3.OA.A.33.MD.D.8 · adapt · grade 3

Archetype: Objects versus Gaps (Fencepost Counting) · step in a 5-type progression

Around the edge of a round pond, $31$ flags are set up at equal intervals of $18\ \text{ft}$ apart. What is the distance around the pond, in feet? (Ignore the thickness of the flags.)

Show solution

Understand

Thirty-one flags stand at equal 18-foot gaps around a round pond. Find the distance all the way around the pond.

Givens

31 flags are placed at equal intervals around the pond.
Each gap between neighboring flags is 18 feet.
The flags are arranged around a closed loop (the pond edge).

Unknowns

The distance around the pond, in feet.

Constraints

The path is a closed loop, so the flags meet up at the start.
Flag thickness is ignored.

Plan

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem

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.

Execute

#9 Solve an Easier Related Problem 3.OA.A.3

Draw a small ring with, say, 4 flags: there are 4 gaps because the last flag connects back to the first. So on a closed loop the number of gaps equals the number of flags. With 31 flags there are 31 gaps.

\text{number of gaps} = \text{number of flags} = 31

On a circle there is no loose end, so each flag starts exactly one gap.

#1 Draw a Diagram 3.MD.D.8

Each of the 31 gaps is 18 feet, so the distance around is 31 times 18.

31 \times 18 = 31 \times 20 - 31 \times 2 = 620 - 62 = 558

Adding up equal gaps around the edge is just multiplication, and the total is the perimeter.

Answer: 558 ft

Review

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.

Compute 31 x 18 directly: 30 x 18 = 540 plus 1 x 18 = 18 gives 558 feet, confirming the loop rule.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Reasoning that gaps equal flags and setting up the multiplication.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the total distance around the closed loop.

💡 On a loop, gaps equal flags -- draw a tiny ring and Grade 3 multiplication does the rest!