Count posts via length divided by spacing

3.OA.A.33.MD.D.8 · adapt · grade 3

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

A straight walking trail is $98$ feet long. Trees are to be planted along both sides of the trail, spaced $7$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

Understand

A straight 98-foot trail has trees planted every 7 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 98 feet long and straight.
Trees are spaced 7 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

Plan

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

Execute

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 7-foot gaps between trees on one side.

98 \div 7 = 14

Each gap is one 7-foot step, so the length divided by the step gives the gap count.

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

14 + 1 = 15

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 15 trees, and there are two sides, so multiply by 2.

15 \times 2 = 30

Both sides are identical, so the total is just twice one side's count.

Answer: 30 trees

Review

15 trees per side over 98 feet (14 gaps of 7 feet = 98 feet) fits exactly, and doubling to 30 for two sides is reasonable. Forgetting the extra end tree would wrongly give 28.

Solve an easier related problem (tool 9): a 14-foot trail with 7-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 98 by 7 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡 This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!