Practice · Find perimeter path length by multiplication · Sensim Math

Variant 1 easy answer: 56 m

Around the edge of a pond, $7$ trees are planted at equal intervals of $8$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Seven trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $8$ m.

Show solution

1 · Understandwhat's really being asked

Seven trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 8 m. We must find the distance all the way around the pond.

Givens

7 trees spaced equally around the pond's edge
The gap between two neighboring trees is 8 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 7 trees make 7 gaps.

7 \text{ trees} \Rightarrow 7 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 7 equal gaps is 8 m, so the distance all the way around is 7 times 8.

7 \times 8 = 56 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 7 copies of 8 m.

Answer: 56 m

4 · Reviewdoes it hold up?

Seven gaps of 8 m total 56 m. If this were a straight line of 7 trees there would be only 6 gaps (48 m), but a closed pond has the same number of gaps as trees, so 56 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 8 + 8 + 8 + 8 + 8 + 8 + 8 = 56 m, which agrees with the multiplication.

Standardsmin 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 7 gaps by 8 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 7 gaps of 8 m means 7 times 8 = 56 m!

Variant 2 easy answer: 48 m

Around the edge of a pond, $8$ trees are planted at equal intervals of $6$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Eight trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $6$ m.

Show solution

1 · Understandwhat's really being asked

Eight trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 6 m. We must find the distance all the way around the pond.

Givens

8 trees spaced equally around the pond's edge
The gap between two neighboring trees is 6 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 8 trees make 8 gaps.

8 \text{ trees} \Rightarrow 8 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 8 equal gaps is 6 m, so the distance all the way around is 8 times 6.

8 \times 6 = 48 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 8 copies of 6 m.

Answer: 48 m

4 · Reviewdoes it hold up?

Eight gaps of 6 m total 48 m. If this were a straight line of 8 trees there would be only 7 gaps (42 m), but a closed pond has the same number of gaps as trees, so 48 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 48 m, which agrees with the multiplication.

Standardsmin 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 8 gaps by 6 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 8 gaps of 6 m means 8 times 6 = 48 m!

Variant 3 easy answer: 45 m

Around the edge of a pond, $5$ trees are planted at equal intervals of $9$ 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 $9$ m.

Show solution

1 · Understandwhat's really being asked

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 9 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 9 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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.

2Multiply gaps by the spacing

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

Each of the 5 equal gaps is 9 m, so the distance all the way around is 5 times 9.

5 \times 9 = 45 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 5 copies of 9 m.

Answer: 45 m

4 · Reviewdoes it hold up?

Five gaps of 9 m total 45 m. If this were a straight line of 5 trees there would be only 4 gaps (36 m), but a closed pond has the same number of gaps as trees, so 45 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 9 + 9 + 9 + 9 + 9 = 45 m, which agrees with the multiplication.

Standardsmin 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 9 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 5 gaps of 9 m means 5 times 9 = 45 m!

Variant 4 medium answer: 60 m

Around the edge of a pond, $6$ trees are planted at equal intervals of $10$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Six trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $10$ m.

Show solution

1 · Understandwhat's really being asked

Six trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 10 m. We must find the distance all the way around the pond.

Givens

6 trees spaced equally around the pond's edge
The gap between two neighboring trees is 10 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 6 trees make 6 gaps.

6 \text{ trees} \Rightarrow 6 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 6 equal gaps is 10 m, so the distance all the way around is 6 times 10.

6 \times 10 = 60 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 6 copies of 10 m.

Answer: 60 m

4 · Reviewdoes it hold up?

Six gaps of 10 m total 60 m. If this were a straight line of 6 trees there would be only 5 gaps (50 m), but a closed pond has the same number of gaps as trees, so 60 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 10 + 10 + 10 + 10 + 10 + 10 = 60 m, which agrees with the multiplication.

Standardsmin 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 6 gaps by 10 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 6 gaps of 10 m means 6 times 10 = 60 m!

Variant 5 medium answer: 66 m

Around the edge of a pond, $6$ trees are planted at equal intervals of $11$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Six trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $11$ m.

Show solution

1 · Understandwhat's really being asked

Six trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 11 m. We must find the distance all the way around the pond.

Givens

6 trees spaced equally around the pond's edge
The gap between two neighboring trees is 11 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 6 trees make 6 gaps.

6 \text{ trees} \Rightarrow 6 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 6 equal gaps is 11 m, so the distance all the way around is 6 times 11.

6 \times 11 = 66 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 6 copies of 11 m.

Answer: 66 m

4 · Reviewdoes it hold up?

Six gaps of 11 m total 66 m. If this were a straight line of 6 trees there would be only 5 gaps (55 m), but a closed pond has the same number of gaps as trees, so 66 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 11 + 11 + 11 + 11 + 11 + 11 = 66 m, which agrees with the multiplication.

Standardsmin 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 6 gaps by 11 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 6 gaps of 11 m means 6 times 11 = 66 m!

Variant 6 medium answer: 48 m

Around the edge of a pond, $4$ trees are planted at equal intervals of $12$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Four trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $12$ m.

Show solution

1 · Understandwhat's really being asked

Four trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 12 m. We must find the distance all the way around the pond.

Givens

4 trees spaced equally around the pond's edge
The gap between two neighboring trees is 12 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 4 trees make 4 gaps.

4 \text{ trees} \Rightarrow 4 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 4 equal gaps is 12 m, so the distance all the way around is 4 times 12.

4 \times 12 = 48 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 4 copies of 12 m.

Answer: 48 m

4 · Reviewdoes it hold up?

Four gaps of 12 m total 48 m. If this were a straight line of 4 trees there would be only 3 gaps (36 m), but a closed pond has the same number of gaps as trees, so 48 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 12 + 12 + 12 + 12 = 48 m, which agrees with the multiplication.

Standardsmin 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 4 gaps by 12 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 4 gaps of 12 m means 4 times 12 = 48 m!

Variant 7 hard answer: 70 m

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

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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.

2Multiply gaps by the spacing

#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

4 · Reviewdoes it hold up?

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.

Another way: Add the equal gaps one at a time around the loop: 14 + 14 + 14 + 14 + 14 = 70 m, which agrees with the multiplication.

Standardsmin 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

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

Variant 8 hard answer: 60 m

Around the edge of a pond, $4$ trees are planted at equal intervals of $15$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Four trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $15$ m.

Show solution

1 · Understandwhat's really being asked

Four trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 15 m. We must find the distance all the way around the pond.

Givens

4 trees spaced equally around the pond's edge
The gap between two neighboring trees is 15 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 4 trees make 4 gaps.

4 \text{ trees} \Rightarrow 4 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 4 equal gaps is 15 m, so the distance all the way around is 4 times 15.

4 \times 15 = 60 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 4 copies of 15 m.

Answer: 60 m

4 · Reviewdoes it hold up?

Four gaps of 15 m total 60 m. If this were a straight line of 4 trees there would be only 3 gaps (45 m), but a closed pond has the same number of gaps as trees, so 60 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 15 + 15 + 15 + 15 = 60 m, which agrees with the multiplication.

Standardsmin 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 4 gaps by 15 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 4 gaps of 15 m means 4 times 15 = 60 m!

Variant 9 hard answer: 60 m

Around the edge of a pond, $3$ trees are planted at equal intervals of $20$ m apart. What is the distance all the way around the pond, in meters?

(Figure) A top-down view of a round (oval) pond. Three trees are planted at equal spacing all the way around the pond's edge, and the gap between two neighboring trees is labeled $20$ m.

Show solution

1 · Understandwhat's really being asked

Three trees are planted at equal spacing all the way around the edge of a round (oval) pond, and each gap between neighboring trees is 20 m. We must find the distance all the way around the pond.

Givens

3 trees spaced equally around the pond's edge
The gap between two neighboring trees is 20 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

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

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.

3 · Execute2 carry out the plan

1Count the gaps on a closed loop

#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: 3 trees make 3 gaps.

3 \text{ trees} \Rightarrow 3 \text{ gaps}

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

2Multiply gaps by the spacing

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

Each of the 3 equal gaps is 20 m, so the distance all the way around is 3 times 20.

3 \times 20 = 60 \text{ m}

Equal gaps mean equal-size jumps, so the total is just 3 copies of 20 m.

Answer: 60 m

4 · Reviewdoes it hold up?

Three gaps of 20 m total 60 m. If this were a straight line of 3 trees there would be only 2 gaps (40 m), but a closed pond has the same number of gaps as trees, so 60 m is correct and a reasonable distance around a pond.

Another way: Add the equal gaps one at a time around the loop: 20 + 20 + 20 = 60 m, which agrees with the multiplication.

Standardsmin 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 3 gaps by 20 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

💡Takeaway. Around a closed loop the gaps equal the trees, so 3 gaps of 20 m means 3 times 20 = 60 m!

Find perimeter path length by multiplication · 9 practice problems

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