Practice · Cuts-to-pieces correspondence rule · Sensim Math

Variant 1 easy answer: △ = ○ + 1; 8 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $7$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 7 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 7 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 7 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 7 cuts

#5 Look for a Pattern 5.OA.A.2

Put 7 in for the cuts.

△ = 7 + 1 = 8

7 cuts give 8 pieces.

Answer: △ = ○ + 1; 8 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 8 pieces need 7 cuts between them, which is the 7 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 7 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 2 easy answer: △ = ○ + 1; 9 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $8$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 8 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 8 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 8 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 8 cuts

#5 Look for a Pattern 5.OA.A.2

Put 8 in for the cuts.

△ = 8 + 1 = 9

8 cuts give 9 pieces.

Answer: △ = ○ + 1; 9 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 9 pieces need 8 cuts between them, which is the 8 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 8 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 3 easy answer: △ = ○ + 1; 10 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $9$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 9 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 9 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 9 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 9 cuts

#5 Look for a Pattern 5.OA.A.2

Put 9 in for the cuts.

△ = 9 + 1 = 10

9 cuts give 10 pieces.

Answer: △ = ○ + 1; 10 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 10 pieces need 9 cuts between them, which is the 9 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 9 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 4 easy answer: △ = ○ + 1; 11 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $10$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 10 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 10 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 10 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 10 cuts

#5 Look for a Pattern 5.OA.A.2

Put 10 in for the cuts.

△ = 10 + 1 = 11

10 cuts give 11 pieces.

Answer: △ = ○ + 1; 11 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 11 pieces need 10 cuts between them, which is the 10 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 10 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 5 medium answer: △ = ○ + 1; 13 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $12$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 12 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 12 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 12 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 12 cuts

#5 Look for a Pattern 5.OA.A.2

Put 12 in for the cuts.

△ = 12 + 1 = 13

12 cuts give 13 pieces.

Answer: △ = ○ + 1; 13 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 13 pieces need 12 cuts between them, which is the 12 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 12 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 6 medium answer: △ = ○ + 1; 15 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $14$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 14 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 14 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 14 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 14 cuts

#5 Look for a Pattern 5.OA.A.2

Put 14 in for the cuts.

△ = 14 + 1 = 15

14 cuts give 15 pieces.

Answer: △ = ○ + 1; 15 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 15 pieces need 14 cuts between them, which is the 14 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 14 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 7 medium answer: △ = ○ + 1; 16 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $15$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 15 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 15 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 15 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 15 cuts

#5 Look for a Pattern 5.OA.A.2

Put 15 in for the cuts.

△ = 15 + 1 = 16

15 cuts give 16 pieces.

Answer: △ = ○ + 1; 16 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 16 pieces need 15 cuts between them, which is the 15 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 15 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 8 medium answer: △ = ○ + 1; 19 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $18$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 18 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 18 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 18 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 18 cuts

#5 Look for a Pattern 5.OA.A.2

Put 18 in for the cuts.

△ = 18 + 1 = 19

18 cuts give 19 pieces.

Answer: △ = ○ + 1; 19 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 19 pieces need 18 cuts between them, which is the 18 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 18 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 9 hard answer: △ = ○ + 1; 21 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $20$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 20 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 20 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 20 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 20 cuts

#5 Look for a Pattern 5.OA.A.2

Put 20 in for the cuts.

△ = 20 + 1 = 21

20 cuts give 21 pieces.

Answer: △ = ○ + 1; 21 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 21 pieces need 20 cuts between them, which is the 20 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 20 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 10 hard answer: △ = ○ + 1; 23 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $22$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 22 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 22 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 22 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 22 cuts

#5 Look for a Pattern 5.OA.A.2

Put 22 in for the cuts.

△ = 22 + 1 = 23

22 cuts give 23 pieces.

Answer: △ = ○ + 1; 23 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 23 pieces need 22 cuts between them, which is the 22 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 22 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 11 hard answer: △ = ○ + 1; 26 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $25$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 25 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 25 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 25 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 25 cuts

#5 Look for a Pattern 5.OA.A.2

Put 25 in for the cuts.

△ = 25 + 1 = 26

25 cuts give 26 pieces.

Answer: △ = ○ + 1; 26 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 26 pieces need 25 cuts between them, which is the 25 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 25 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Variant 12 hard answer: △ = ○ + 1; 31 pieces

A rope is cut to divide it into several pieces. Let the number of cuts be $○$ and the number of pieces be $△$ . Write an equation that shows the correspondence between $○$ and $△$ , and find how many pieces the rope is divided into when it is cut $30$ times.

Show solution

1 · Understandwhat's really being asked

Cutting a rope makes pieces. We must write the rule connecting the number of cuts to the number of pieces, and then use it for 30 cuts.

Givens

The rope is cut straight across, into separate pieces.
The number of cuts is called ○ and the number of pieces △.

Unknowns

The rule linking ○ and △, and the pieces after 30 cuts.

Constraints

Every cut goes all the way through, and no piece is cut twice at once.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#2 Make a Systematic List

Ten cuts is hard to picture, one cut is not. Try the small cases, list what happens, and the rule shows itself -- then it answers 30 cuts without any picturing at all.

3 · Execute4 carry out the plan

1Try small numbers of cuts

#9 Solve an Easier Related Problem 4.OA.C.5

One cut splits the rope in two. A second cut splits one of those, making three. A third makes four.

1 \to 2,\quad 2 \to 3,\quad 3 \to 4

Each new cut adds exactly one more piece.

2List the pairs and spot the pattern

#2 Make a Systematic List 5.OA.B.3

Writing cuts and pieces side by side makes the gap between the two columns plain.

(1, 2), (2, 3), (3, 4), (4, 5)

The pieces are always one ahead.

3Write the correspondence as an equation

#5 Look for a Pattern 5.OA.A.2

The uncut rope is already one piece, and every cut adds another -- so pieces are cuts plus that original one.

△ = ○ + 1

The 'plus one' is the rope you started with.

4Use the rule for 30 cuts

#5 Look for a Pattern 5.OA.A.2

Put 30 in for the cuts.

△ = 30 + 1 = 31

30 cuts give 31 pieces.

Answer: △ = ○ + 1; 31 pieces

4 · Reviewdoes it hold up?

Count backwards from the answer: 31 pieces need 30 cuts between them, which is the 30 we were given.

Another way: Think of the cuts as the gaps between pieces laid in a row: n pieces in a line have n - 1 gaps, the same relationship read the other way.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Generating the small cases to see the pattern.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the cut count with the piece count.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 30 cuts.

💡Takeaway. Pieces run one ahead of cuts, because the rope was already one piece before you started.

Cuts-to-pieces correspondence rule · 12 practice problems

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