Practice · Objects versus gaps when spacing along a line · Sensim Math

Variant 1 easy

Trees are planted along one side of a straight road that is $10$ m long, from the very beginning to the very end, spaced $2$ 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 $10$ 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 $2$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 10 m long, trees are planted from the very start to the very end, spaced 2 m apart. I need to find how many trees there are in all.

Givens

The road is 10 m long.
Trees are spaced 2 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 (2 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 2 m. Divide the road length by the gap length to find how many gaps there are.

10 \div 2 = 5

Fitting equal lengths into a total length is a division: how many 2 m pieces make 10 m.

2Turn gaps into trees using the diagram

#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 = 6

Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.

Answer: 6 trees

4 · Reviewdoes it hold up?

The figure shows 6 trees and 5 gaps, matching the count. Checking the length: 5 gaps of 2 m each is 5 times 2 equals 10 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 5 gaps give 6 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 2 easy

Trees are planted along one side of a straight road that is $12$ m long, from the very beginning to the very end, spaced $3$ 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 $12$ 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 $3$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 12 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 12 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).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#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.

12 \div 3 = 4

Fitting equal lengths into a total length is a division: how many 3 m pieces make 12 m.

2Turn gaps into trees using the diagram

#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.

4 + 1 = 5

Picture fence posts: 4 spans between posts means 5 posts, because the two endpoints each carry a post.

Answer: 5 trees

4 · Reviewdoes it hold up?

The figure shows 5 trees and 4 gaps, matching the count. Checking the length: 4 gaps of 3 m each is 4 times 3 equals 12 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 4 gaps give 5 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 3 easy

Trees are planted along one side of a straight road that is $15$ m long, from the very beginning to the very end, spaced $3$ 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 $15$ 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 $3$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#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 \div 3 = 5

Fitting equal lengths into a total length is a division: how many 3 m pieces make 15 m.

2Turn gaps into trees using the diagram

#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 = 6

Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.

Answer: 6 trees

4 · Reviewdoes it hold up?

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.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 5 gaps give 6 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 4 medium

Trees are planted along one side of a straight road that is $18$ m long, from the very beginning to the very end, spaced $2$ 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 $18$ 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 $2$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 18 m long, trees are planted from the very start to the very end, spaced 2 m apart. I need to find how many trees there are in all.

Givens

The road is 18 m long.
Trees are spaced 2 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 (2 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 2 m. Divide the road length by the gap length to find how many gaps there are.

18 \div 2 = 9

Fitting equal lengths into a total length is a division: how many 2 m pieces make 18 m.

2Turn gaps into trees using the diagram

#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.

9 + 1 = 10

Picture fence posts: 9 spans between posts means 10 posts, because the two endpoints each carry a post.

Answer: 10 trees

4 · Reviewdoes it hold up?

The figure shows 10 trees and 9 gaps, matching the count. Checking the length: 9 gaps of 2 m each is 9 times 2 equals 18 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 9 gaps give 10 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 5 medium

Trees are planted along one side of a straight road that is $20$ m long, from the very beginning to the very end, spaced $4$ 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 $20$ 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 $4$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 20 m long, trees are planted from the very start to the very end, spaced 4 m apart. I need to find how many trees there are in all.

Givens

The road is 20 m long.
Trees are spaced 4 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 (4 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 4 m. Divide the road length by the gap length to find how many gaps there are.

20 \div 4 = 5

Fitting equal lengths into a total length is a division: how many 4 m pieces make 20 m.

2Turn gaps into trees using the diagram

#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 = 6

Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.

Answer: 6 trees

4 · Reviewdoes it hold up?

The figure shows 6 trees and 5 gaps, matching the count. Checking the length: 5 gaps of 4 m each is 5 times 4 equals 20 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 5 gaps give 6 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 6 medium

Trees are planted along one side of a straight road that is $24$ m long, from the very beginning to the very end, spaced $6$ 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 $24$ 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 $6$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 24 m long, trees are planted from the very start to the very end, spaced 6 m apart. I need to find how many trees there are in all.

Givens

The road is 24 m long.
Trees are spaced 6 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 (6 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 6 m. Divide the road length by the gap length to find how many gaps there are.

24 \div 6 = 4

Fitting equal lengths into a total length is a division: how many 6 m pieces make 24 m.

2Turn gaps into trees using the diagram

#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.

4 + 1 = 5

Picture fence posts: 4 spans between posts means 5 posts, because the two endpoints each carry a post.

Answer: 5 trees

4 · Reviewdoes it hold up?

The figure shows 5 trees and 4 gaps, matching the count. Checking the length: 4 gaps of 6 m each is 4 times 6 equals 24 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 4 gaps give 5 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 7 hard

Trees are planted along one side of a straight road that is $30$ m long, from the very beginning to the very end, spaced $5$ 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 $30$ 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 $5$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 30 m long, trees are planted from the very start to the very end, spaced 5 m apart. I need to find how many trees there are in all.

Givens

The road is 30 m long.
Trees are spaced 5 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 (5 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 5 m. Divide the road length by the gap length to find how many gaps there are.

30 \div 5 = 6

Fitting equal lengths into a total length is a division: how many 5 m pieces make 30 m.

2Turn gaps into trees using the diagram

#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.

6 + 1 = 7

Picture fence posts: 6 spans between posts means 7 posts, because the two endpoints each carry a post.

Answer: 7 trees

4 · Reviewdoes it hold up?

The figure shows 7 trees and 6 gaps, matching the count. Checking the length: 6 gaps of 5 m each is 6 times 5 equals 30 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 6 gaps give 7 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 8 hard

Trees are planted along one side of a straight road that is $36$ m long, from the very beginning to the very end, spaced $9$ 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 $36$ 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 $9$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 36 m long, trees are planted from the very start to the very end, spaced 9 m apart. I need to find how many trees there are in all.

Givens

The road is 36 m long.
Trees are spaced 9 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 (9 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 9 m. Divide the road length by the gap length to find how many gaps there are.

36 \div 9 = 4

Fitting equal lengths into a total length is a division: how many 9 m pieces make 36 m.

2Turn gaps into trees using the diagram

#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.

4 + 1 = 5

Picture fence posts: 4 spans between posts means 5 posts, because the two endpoints each carry a post.

Answer: 5 trees

4 · Reviewdoes it hold up?

The figure shows 5 trees and 4 gaps, matching the count. Checking the length: 4 gaps of 9 m each is 4 times 9 equals 36 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 4 gaps give 5 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Variant 9 hard

Trees are planted along one side of a straight road that is $40$ m long, from the very beginning to the very end, spaced $8$ 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 $40$ 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 $8$ m.

Your answer

Show solution

1 · Understandwhat's really being asked

Along one side of a straight road 40 m long, trees are planted from the very start to the very end, spaced 8 m apart. I need to find how many trees there are in all.

Givens

The road is 40 m long.
Trees are spaced 8 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 (8 m).

2 · Planchoose the strategy

#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 equal gaps fit in the road (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.

3 · Execute2 carry out the plan

1Count the equal gaps along the road

#7 Identify Subproblems 3.OA.A.3

Each gap between neighboring trees is 8 m. Divide the road length by the gap length to find how many gaps there are.

40 \div 8 = 5

Fitting equal lengths into a total length is a division: how many 8 m pieces make 40 m.

2Turn gaps into trees using the diagram

#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 = 6

Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.

Answer: 6 trees

4 · Reviewdoes it hold up?

The figure shows 6 trees and 5 gaps, matching the count. Checking the length: 5 gaps of 8 m each is 5 times 8 equals 40 m, exactly the road length, so the trees reach from start to end with none left over.

Another way: Solve an easier related problem: a one-gap road has 2 trees, a two-gap road has 3 trees; the pattern "trees = gaps + 1" makes 5 gaps give 6 trees.

Standardsmin 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.

💡Takeaway. Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!

Objects versus gaps when spacing along a line · 9 practice problems

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