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← 2-2 · Gaps are one fewer than the objects · Objects versus Gaps (Fencepost Counting)

Gaps are one fewer than the objects · 8 practice problems

2.MD.B.5

Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 4 gaps

55 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 5 gap gap gap gap
Show solution

Understand

5 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 5 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 5 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4, post4-post5.
5 posts→spaces between neighbors5 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 5 posts, subtract 1 to get the number of gaps.
5−1=45 - 1 = 4
Subtracting one matches the 4 spaces we drew between the 5 posts.
Answer: 4 gaps

Review

We can directly count the 4 spaces between the 5 posts in the picture, so 4 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4, 4-5) and count the entries, giving 4.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 2 answer: 5 gaps

66 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 5 6 gap gap gap gap gap
Show solution

Understand

6 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 6 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 6 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4, post4-post5, post5-post6.
6 posts→spaces between neighbors6 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 6 posts, subtract 1 to get the number of gaps.
6−1=56 - 1 = 5
Subtracting one matches the 5 spaces we drew between the 6 posts.
Answer: 5 gaps

Review

We can directly count the 5 spaces between the 6 posts in the picture, so 5 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4, 4-5, 5-6) and count the entries, giving 5.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 3 answer: 1 gaps

22 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 gap
Show solution

Understand

2 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 2 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 2 posts in a line, the spaces appear only between neighbors: post1-post2.
2 posts→spaces between neighbors2 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 2 posts, subtract 1 to get the number of gaps.
2−1=12 - 1 = 1
Subtracting one matches the 1 spaces we drew between the 2 posts.
Answer: 1 gaps

Review

We can directly count the 1 spaces between the 2 posts in the picture, so 1 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2) and count the entries, giving 1.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 4 answer: 3 gaps

44 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 gap gap gap
Show solution

Understand

4 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 4 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 4 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4.
4 posts→spaces between neighbors4 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 4 posts, subtract 1 to get the number of gaps.
4−1=34 - 1 = 3
Subtracting one matches the 3 spaces we drew between the 4 posts.
Answer: 3 gaps

Review

We can directly count the 3 spaces between the 4 posts in the picture, so 3 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4) and count the entries, giving 3.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 5 answer: 7 gaps

88 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 5 6 7 8 gap gap gap gap gap gap gap
Show solution

Understand

8 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 8 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 8 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4, post4-post5, post5-post6, post6-post7, post7-post8.
8 posts→spaces between neighbors8 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 8 posts, subtract 1 to get the number of gaps.
8−1=78 - 1 = 7
Subtracting one matches the 7 spaces we drew between the 8 posts.
Answer: 7 gaps

Review

We can directly count the 7 spaces between the 8 posts in the picture, so 7 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4, 4-5, 5-6, 6-7, 7-8) and count the entries, giving 7.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 6 answer: 8 gaps

99 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 5 6 7 8 9 gap gap gap gap gap gap gap gap
Show solution

Understand

9 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 9 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 9 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4, post4-post5, post5-post6, post6-post7, post7-post8, post8-post9.
9 posts→spaces between neighbors9 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 9 posts, subtract 1 to get the number of gaps.
9−1=89 - 1 = 8
Subtracting one matches the 8 spaces we drew between the 9 posts.
Answer: 8 gaps

Review

We can directly count the 8 spaces between the 9 posts in the picture, so 8 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4, 4-5, 5-6, 6-7, 7-8, 8-9) and count the entries, giving 8.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 7 answer: 6 gaps

77 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 4 5 6 7 gap gap gap gap gap gap
Show solution

Understand

7 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 7 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 7 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3, post3-post4, post4-post5, post5-post6, post6-post7.
7 posts→spaces between neighbors7 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 7 posts, subtract 1 to get the number of gaps.
7−1=67 - 1 = 6
Subtracting one matches the 6 spaces we drew between the 7 posts.
Answer: 6 gaps

Review

We can directly count the 6 spaces between the 7 posts in the picture, so 6 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3, 3-4, 4-5, 5-6, 6-7) and count the entries, giving 6.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!
Variant 8 answer: 2 gaps

33 wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

1 2 3 gap gap
Show solution

Understand

3 posts stand in a row, evenly spaced. We need to count how many gaps (spaces between neighboring posts) there are.

Givens
  • There are 3 posts in a straight line.
  • The posts are equally spaced.
  • A gap is the space between two neighboring posts.
Unknowns
  • The total number of gaps between the posts.
Constraints
  • Gaps only exist between two posts, not before the first or after the last post.

Plan

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

Draw the row of posts and mark each space. Each gap sits between two posts, so the count of gaps is always one fewer than the count of posts. A quick small-case pattern confirms the rule.

Execute

#1 Draw a Diagram 2.MD.B.5
Drawing 3 posts in a line, the spaces appear only between neighbors: post1-post2, post2-post3.
3 posts→spaces between neighbors3 \text{ posts} \rightarrow \text{spaces between neighbors}
Seeing the line of posts makes it clear the ends have no gap beyond them.
#5 Look for a Pattern 2.MD.B.5
2 posts make 1 gap, 3 posts make 2 gaps, 4 posts make 3 gaps. So the number of gaps is the number of posts minus 1.
gaps=posts−1\text{gaps} = \text{posts} - 1
A repeated small-case pattern reveals the simple rule without listing every gap.
#1 Draw a Diagram 2.MD.B.5
With 3 posts, subtract 1 to get the number of gaps.
3−1=23 - 1 = 2
Subtracting one matches the 2 spaces we drew between the 3 posts.
Answer: 2 gaps

Review

We can directly count the 2 spaces between the 3 posts in the picture, so 2 is correct and the unit (gaps, a count) matches the question.

Make a Systematic List (tool 2): list each adjacent pair (1-2, 2-3) and count the entries, giving 2.

Standards · min grade 2

  • 2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.
💡 Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!