← 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

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Generated variants — 8

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

Variant 1 easy

$2$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#1 Draw a Diagram 2.MD.B.5

Drawing 2 posts in a line, the spaces appear only between neighbors: post1-post2.

2 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 2 posts, subtract 1 to get the number of gaps.

2 - 1 = 1

Subtracting one matches the 1 spaces we drew between the 2 posts.

Answer: 1 gaps

4 · Reviewdoes it hold up?

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.

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

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 2 easy

$3$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 3 posts, subtract 1 to get the number of gaps.

3 - 1 = 2

Subtracting one matches the 2 spaces we drew between the 3 posts.

Answer: 2 gaps

4 · Reviewdoes it hold up?

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.

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

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 3 medium

$4$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 4 posts, subtract 1 to get the number of gaps.

4 - 1 = 3

Subtracting one matches the 3 spaces we drew between the 4 posts.

Answer: 3 gaps

4 · Reviewdoes it hold up?

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.

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

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 4 medium

$5$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 5 posts, subtract 1 to get the number of gaps.

5 - 1 = 4

Subtracting one matches the 4 spaces we drew between the 5 posts.

Answer: 4 gaps

4 · Reviewdoes it hold up?

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.

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

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 5 medium

$6$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 6 posts, subtract 1 to get the number of gaps.

6 - 1 = 5

Subtracting one matches the 5 spaces we drew between the 6 posts.

Answer: 5 gaps

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 6 medium

$7$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 7 posts, subtract 1 to get the number of gaps.

7 - 1 = 6

Subtracting one matches the 6 spaces we drew between the 7 posts.

Answer: 6 gaps

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 7 hard

$8$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 8 posts, subtract 1 to get the number of gaps.

8 - 1 = 7

Subtracting one matches the 7 spaces we drew between the 8 posts.

Answer: 7 gaps

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

Variant 8 hard

$9$ wooden posts are placed in a row at equal spacing.

Find the total number of gaps between the wooden posts.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Picture the posts and spaces

#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 \text{ posts} \rightarrow \text{spaces between neighbors}

Seeing the line of posts makes it clear the ends have no gap beyond them.

2Spot the pattern: gaps are one fewer

#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.

\text{gaps} = \text{posts} - 1

A repeated small-case pattern reveals the simple rule without listing every gap.

3Apply the rule

#1 Draw a Diagram 2.MD.B.5

With 9 posts, subtract 1 to get the number of gaps.

9 - 1 = 8

Subtracting one matches the 8 spaces we drew between the 9 posts.

Answer: 8 gaps

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Reasoning about spaces between equally spaced objects along a line.

💡Takeaway. Spaces are always one fewer than the posts -- Grade 2 counting sense you can see in the picture!

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