Intervals differ on open paths versus closed loops

4.OA.A.34.MD.A.2 · adapt · grade 4

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

Trees are to be planted along a straight road that is $200$ m long, spaced $20$ m apart from one end of the road all the way to the other end. How many trees are needed in all?

(Figure) A straight road with both ends open. The gap between the left end of the road and the next tree is labeled $20$ m, and the total length of the road is labeled $200$ m underneath. A tree stands at each end of the road, and the middle stretch is shown as "……" to indicate that the trees continue at the same regular spacing.

Show solution

Understand

Along a straight 200 m road, trees are planted every 20 m starting at one end and continuing to the other end, with a tree at each end. Find the total number of trees.

Givens

The road is straight and 200 m long
Trees are spaced 20 m apart
A tree stands at each end of the road (the figure shows a tree at the left end and the right end, with evenly spaced trees in between)
The road is open (not a loop)

Unknowns

The total number of trees needed

Constraints

Spacing is uniform at 20 m
On an open straight road, the number of trees is one more than the number of gaps

Plan

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

Interval-counting is best seen with a diagram: the figure shows trees at both ends of an open road, so the count of trees is the count of gaps plus 1. Trying a tiny case (a short road with a couple of gaps) makes the plus-one rule obvious before applying it to 200 m.

Execute

#9 Solve an Easier Related Problem 4.OA.A.3

Divide the road length by the spacing to find how many 20 m gaps fit: 200 / 20 = 10 gaps.

200 \div 20 = 10 \text{ gaps}

Each 20 m piece is one gap, so dividing the total length by 20 counts them.

#1 Draw a Diagram 4.MD.A.2

On a straight open road with a tree at both ends, the number of trees is one more than the number of gaps because both endpoints get a tree. So trees = 10 + 1 = 11. (The figure confirms a tree sits at each end.)

10 + 1 = 11 \text{ trees}

Drawing posts and the gaps between them shows there is always one more post than gap on an open path.

Answer: 11 trees

Review

Check with the spacing: 11 trees make 10 gaps of 20 m, which is 10 x 20 = 200 m, exactly the road length. The plus-one matches the figure showing a tree at each end.

Solve an easier related problem (tool 9): a 40 m road every 20 m gives gaps at 0, 20, 40 -- that is 2 gaps but 3 trees, confirming the trees-equals-gaps-plus-1 rule, then scale up to 200 m.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Dividing 200 by 20 to count the gaps and adding one for the endpoint tree.
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money — Reasoning about lengths and spacing along the road to relate gaps to trees.

💡 This only needs Grade 4 dividing -- count the gaps, then remember an open road has one more tree than gaps!