Problem
Count posts via length divided by spacing
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.
OperationsMeasurement & data
Your answer
Watch out It's easy to stop at the gap count alone (14 per side) — the trail's own two ends both get a tree, so trees = gaps + 1.
How to solve
Strategy Draw a Diagram — 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.
1STEP 1
Count the gaps along one side
Dividing the trail length by the spacing gives the number of 7-foot gaps between trees on one side.
98 ÷ 7 = 14
Each gap is one 7-foot step, so the length divided by the step gives the gap count.
3.OA.C.7Draw A Diagram2STEP 2
Add one for the posts-versus-gaps rule
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.
3.OA.A.3Look For A PatternOn a straight trail with a tree planted at both ends, the number of trees is one more than the number of equal gaps between them.
Why?
One tree stands at the very start before any gap, and every gap after it is closed by exactly one more tree, so the count is that first tree plus one tree for each gap.
Why?
Each gap is matched to the single tree that closes it, and no two gaps close at the same tree, so the gaps and the trees after the start pair up exactly one to one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The whole line of trees splits with no gaps or overlaps into the single starting tree and the trees that close a gap, so the total is one plus the gap count.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Double for both sides
Each side of the trail has 15 trees, and there are two sides, so multiply by 2.
15 × 2 = 30
Both sides are identical, so the total is just twice one side's count.
3.OA.A.3Draw A DiagramAnswer
30
15 × 2 = 30
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.
Takeaway
This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!
- Count the gaps along one side
- Add one for the posts-versus-gaps rule
- Double for both sides
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