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Cut a rectangle into a grid of cards · 8 practice problems

2.G.A.23.MD.C.7

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Generated variants — 8

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

Variant 1 easy

A rectangular sheet of paper is $15\ \text{cm}$ wide and $9\ \text{cm}$ tall. It is cut into identical squares that are each $3\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 15 cm by 9 cm rectangular sheet is cut into identical 3 cm squares. We need the total number of squares.

Givens

The sheet is 15 cm wide and 9 cm tall.
Each square is 3 cm on a side.
The figure shows the sheet divided into a 5-by-3 grid of 3 cm squares.

Unknowns

The total number of identical 3 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (3 divides both 15 and 9 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 15 cm and each square is 3 cm wide, so divide to find the columns.

15 \div 3 = 5

Partitioning a length into equal 3 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 9 cm and each square is 3 cm tall, so divide to find the rows.

9 \div 3 = 3

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 5-column by 3-row array, so multiply to count them all.

5 \times 3 = 15

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 15 squares

4 · Reviewdoes it hold up?

The figure shows exactly 5 columns and 3 rows; counting the grid cells gives 15, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 15x9 = 135 cm^2, each square 3x3 = 9 cm^2, and 135 / 9 = 15.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 3 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 2 easy

A rectangular sheet of paper is $16\ \text{cm}$ wide and $16\ \text{cm}$ tall. It is cut into identical squares that are each $4\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 16 cm by 16 cm rectangular sheet is cut into identical 4 cm squares. We need the total number of squares.

Givens

The sheet is 16 cm wide and 16 cm tall.
Each square is 4 cm on a side.
The figure shows the sheet divided into a 4-by-4 grid of 4 cm squares.

Unknowns

The total number of identical 4 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (4 divides both 16 and 16 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 16 cm and each square is 4 cm wide, so divide to find the columns.

16 \div 4 = 4

Partitioning a length into equal 4 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 16 cm and each square is 4 cm tall, so divide to find the rows.

16 \div 4 = 4

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 4-column by 4-row array, so multiply to count them all.

4 \times 4 = 16

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 16 squares

4 · Reviewdoes it hold up?

The figure shows exactly 4 columns and 4 rows; counting the grid cells gives 16, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 16x16 = 256 cm^2, each square 4x4 = 16 cm^2, and 256 / 16 = 16.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 4 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 3 medium

A rectangular sheet of paper is $18\ \text{cm}$ wide and $12\ \text{cm}$ tall. It is cut into identical squares that are each $6\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 18 cm by 12 cm rectangular sheet is cut into identical 6 cm squares. We need the total number of squares.

Givens

The sheet is 18 cm wide and 12 cm tall.
Each square is 6 cm on a side.
The figure shows the sheet divided into a 3-by-2 grid of 6 cm squares.

Unknowns

The total number of identical 6 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (6 divides both 18 and 12 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 18 cm and each square is 6 cm wide, so divide to find the columns.

18 \div 6 = 3

Partitioning a length into equal 6 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 12 cm and each square is 6 cm tall, so divide to find the rows.

12 \div 6 = 2

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 3-column by 2-row array, so multiply to count them all.

3 \times 2 = 6

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 6 squares

4 · Reviewdoes it hold up?

The figure shows exactly 3 columns and 2 rows; counting the grid cells gives 6, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 18x12 = 216 cm^2, each square 6x6 = 36 cm^2, and 216 / 36 = 6.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 6 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 4 medium

A rectangular sheet of paper is $21\ \text{cm}$ wide and $14\ \text{cm}$ tall. It is cut into identical squares that are each $7\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 21 cm by 14 cm rectangular sheet is cut into identical 7 cm squares. We need the total number of squares.

Givens

The sheet is 21 cm wide and 14 cm tall.
Each square is 7 cm on a side.
The figure shows the sheet divided into a 3-by-2 grid of 7 cm squares.

Unknowns

The total number of identical 7 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (7 divides both 21 and 14 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 21 cm and each square is 7 cm wide, so divide to find the columns.

21 \div 7 = 3

Partitioning a length into equal 7 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 14 cm and each square is 7 cm tall, so divide to find the rows.

14 \div 7 = 2

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 3-column by 2-row array, so multiply to count them all.

3 \times 2 = 6

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 6 squares

4 · Reviewdoes it hold up?

The figure shows exactly 3 columns and 2 rows; counting the grid cells gives 6, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 21x14 = 294 cm^2, each square 7x7 = 49 cm^2, and 294 / 49 = 6.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 7 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 5 medium

A rectangular sheet of paper is $24\ \text{cm}$ wide and $16\ \text{cm}$ tall. It is cut into identical squares that are each $4\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 24 cm by 16 cm rectangular sheet is cut into identical 4 cm squares. We need the total number of squares.

Givens

The sheet is 24 cm wide and 16 cm tall.
Each square is 4 cm on a side.
The figure shows the sheet divided into a 6-by-4 grid of 4 cm squares.

Unknowns

The total number of identical 4 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (4 divides both 24 and 16 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 24 cm and each square is 4 cm wide, so divide to find the columns.

24 \div 4 = 6

Partitioning a length into equal 4 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 16 cm and each square is 4 cm tall, so divide to find the rows.

16 \div 4 = 4

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 6-column by 4-row array, so multiply to count them all.

6 \times 4 = 24

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 24 squares

4 · Reviewdoes it hold up?

The figure shows exactly 6 columns and 4 rows; counting the grid cells gives 24, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 24x16 = 384 cm^2, each square 4x4 = 16 cm^2, and 384 / 16 = 24.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 4 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 6 medium

A rectangular sheet of paper is $30\ \text{cm}$ wide and $20\ \text{cm}$ tall. It is cut into identical squares that are each $5\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 30 cm by 20 cm rectangular sheet is cut into identical 5 cm squares. We need the total number of squares.

Givens

The sheet is 30 cm wide and 20 cm tall.
Each square is 5 cm on a side.
The figure shows the sheet divided into a 6-by-4 grid of 5 cm squares.

Unknowns

The total number of identical 5 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (5 divides both 30 and 20 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 30 cm and each square is 5 cm wide, so divide to find the columns.

30 \div 5 = 6

Partitioning a length into equal 5 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 20 cm and each square is 5 cm tall, so divide to find the rows.

20 \div 5 = 4

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 6-column by 4-row array, so multiply to count them all.

6 \times 4 = 24

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 24 squares

4 · Reviewdoes it hold up?

The figure shows exactly 6 columns and 4 rows; counting the grid cells gives 24, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 30x20 = 600 cm^2, each square 5x5 = 25 cm^2, and 600 / 25 = 24.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 5 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 7 hard

A rectangular sheet of paper is $36\ \text{cm}$ wide and $24\ \text{cm}$ tall. It is cut into identical squares that are each $12\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 36 cm by 24 cm rectangular sheet is cut into identical 12 cm squares. We need the total number of squares.

Givens

The sheet is 36 cm wide and 24 cm tall.
Each square is 12 cm on a side.
The figure shows the sheet divided into a 3-by-2 grid of 12 cm squares.

Unknowns

The total number of identical 12 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (12 divides both 36 and 24 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 36 cm and each square is 12 cm wide, so divide to find the columns.

36 \div 12 = 3

Partitioning a length into equal 12 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 24 cm and each square is 12 cm tall, so divide to find the rows.

24 \div 12 = 2

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 3-column by 2-row array, so multiply to count them all.

3 \times 2 = 6

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 6 squares

4 · Reviewdoes it hold up?

The figure shows exactly 3 columns and 2 rows; counting the grid cells gives 6, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 36x24 = 864 cm^2, each square 12x12 = 144 cm^2, and 864 / 144 = 6.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 12 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Variant 8 hard

A rectangular sheet of paper is $40\ \text{cm}$ wide and $30\ \text{cm}$ tall. It is cut into identical squares that are each $10\ \text{cm}$ on a side.

Find the total number of identical squares that can be made.

Your answer

Show solution

1 · Understandwhat's really being asked

A 40 cm by 30 cm rectangular sheet is cut into identical 10 cm squares. We need the total number of squares.

Givens

The sheet is 40 cm wide and 30 cm tall.
Each square is 10 cm on a side.
The figure shows the sheet divided into a 4-by-3 grid of 10 cm squares.

Unknowns

The total number of identical 10 cm squares cut from the sheet.

Constraints

Squares tile the rectangle exactly (10 divides both 40 and 30 evenly), with no leftover.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into two subproblems: how many squares fit across the width, and how many fit down the height. Then a rows-times-columns array gives the total, matching the grid in the figure.

3 · Execute3 carry out the plan

1Squares across the width

#7 Identify Subproblems 2.G.A.2

The width is 40 cm and each square is 10 cm wide, so divide to find the columns.

40 \div 10 = 4

Partitioning a length into equal 10 cm pieces is a Grade 2 measurement idea.

2Squares down the height

#7 Identify Subproblems 2.G.A.2

The height is 30 cm and each square is 10 cm tall, so divide to find the rows.

30 \div 10 = 3

Same partitioning idea applied to the vertical side gives the number of rows.

3Multiply rows by columns

#7 Identify Subproblems 3.MD.C.7

The squares form a 4-column by 3-row array, so multiply to count them all.

4 \times 3 = 12

An array of equal squares is counted by multiplication, just like finding area in unit squares.

Answer: 12 squares

4 · Reviewdoes it hold up?

The figure shows exactly 4 columns and 3 rows; counting the grid cells gives 12, confirming the array. The unit (squares, a count) matches the question.

Another way: Use area (tool 8, Analyze Units): total area 40x30 = 1200 cm^2, each square 10x10 = 100 cm^2, and 1200 / 100 = 12.

Standardsmin grade 3

2.G.A.2 Partition a rectangle into rows and columns of same-size squares — Finding how many 10 cm squares fit along the width and height.
3.MD.C.7 Relate area to multiplication and addition operations — Counting the full grid of squares as a rows-by-columns array.

💡Takeaway. Count squares across and down, then multiply - the same row-by-column array thinking behind area!

Where next?

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