Find perimeter path length by multiplication

3.MD.D.83.OA.A.3 · adapt · grade 3

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

Around the edge of a pond, $5$ trees are planted at equal intervals of $14$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Five trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $14$ m.

Show solution

Understand

Five trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 14 m. We must find the distance all the way around the pond.

Givens

5 trees spaced equally around the pond's edge
The gap between two neighboring trees is 14 m
The trees go all the way around a closed loop (the figure shows an oval pond)

Unknowns

The total distance around the pond in meters

Constraints

The path is a closed loop, so the trees form a circle of gaps with no open ends

Plan

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

The figure shows trees around a closed loop. On a closed loop the number of gaps equals the number of trees (unlike a straight line), so the perimeter is simply the number of gaps times the gap length.

Execute

#1 Draw a Diagram 3.OA.D.9

Because the trees go all the way around and the path closes back to the start, the number of gaps between trees equals the number of trees: 5 trees make 5 gaps.

5 \text{ trees} \Rightarrow 5 \text{ gaps}

On a circle every tree has a gap leading to the next, with no leftover end, so gaps match trees.

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

Each of the 5 equal gaps is 14 m, so the distance all the way around is 5 times 14.

5 \times 14 = 70 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 5 copies of 14 m.

Answer: 70 m

Review

Five gaps of 14 m total 70 m. If this were a straight line of 5 trees there would be only 4 gaps (56 m), but a closed pond has the same number of gaps as trees, so 70 m is correct and a reasonable distance around a pond.

Add the equal gaps one at a time around the loop (tool 7, subproblems): 14 + 14 + 14 + 14 + 14 = 70 m, which agrees with the multiplication.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that on a closed loop the number of gaps equals the number of trees
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 5 gaps by 14 m to get the total distance
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the perimeter (distance around) of the pond from equal spacings

💡 Around a closed loop the gaps equal the trees, so 5 gaps of 14 m means 5 times 14 = 70 m!