Sensim Math · Depth 한국어

3-1 · Division

Objects versus gaps when spacing along a line

3.OA.A.33.OA.D.8 · adapt · grade 3

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

▶ Practice — 9 problems

Trees are planted along one side of a straight road that is 1515 m long, from the very beginning to the very end, spaced 33 m apart. How many trees are needed in all?

(Figure) A side view of a single straight road. The full length of the road is labeled 1515 m, and trees are planted at equal spacing along one side from the start to the end. The gap between two neighboring trees is labeled 33 m.

3 m 15 m
Show solution

Understand

Along one side of a straight road 15 m long, trees are planted from the very start to the very end, spaced 3 m apart. I need to find how many trees there are in all.

Givens
  • The road is 15 m long.
  • Trees are spaced 3 m apart.
  • A tree is planted at the very beginning and at the very end of the road.
  • The figure shows trees evenly spaced along one side from start to end.
Unknowns
  • The total number of trees planted.
Constraints
  • Trees are planted along one side only.
  • Both ends of the road have a tree.
  • The spacing is equal everywhere (3 m).

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Spacing-along-a-line problems are clearest with a diagram: drawing the road and marking trees shows that the number of gaps and the number of trees are not the same. First find how many 3 m gaps fit in 15 m (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

Execute

#7 Identify Subproblems 3.OA.A.3
Each gap between neighboring trees is 3 m. Divide the road length by the gap length to find how many gaps there are.
15÷3=515 \div 3 = 5
Fitting equal lengths into a total length is a division: how many 3 m pieces make 15 m.
#1 Draw a Diagram 3.OA.D.8
On a straight road with a tree at both the start and the end, the trees are the marks at the ends of the gaps, so there is always one more tree than the number of gaps.
5+1=65 + 1 = 6
Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.
Answer: 6 trees

Review

The figure shows 6 trees and 5 gaps, matching the count. Checking the length: 5 gaps of 3 m each is 5 times 3 equals 15 m, exactly the road length, so the trees reach from start to end with none left over.

Solve an easier related problem (tool 9): a 3 m road has 1 gap and 2 trees, a 6 m road has 2 gaps and 3 trees; the pattern "trees = gaps + 1" makes 5 gaps give 6 trees.

Standards · min grade 3

  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the road length by the spacing to count the equal gaps.
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Adding 1 to the number of gaps to get the number of trees in this two-step problem.
💡 Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!