Practice · Rectangle sides are multiples of the diameter · Sensim Math

Variant 1 easy answer: 9

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the square of side length $6\ \text{cm}$ on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 6 cm by 6 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 6 cm wide and 6 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 6 cm fits 6 / 2 = 3 cells, and down 6 cm fits 6 / 2 = 3 rows of cells.

6 \div 2 = 3,\ 6 \div 2 = 3

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 3 by 3 grid of cells holds one circle per cell.

3 \times 3 = 9

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 9

4 · Reviewdoes it hold up?

9 circles is a whole number, as it must be for counting objects. The 3 columns times 3 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 3 x 3 = 9 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 2 easy answer: 4

You want to draw as many circles with radius $2\ \text{cm}$ as possible inside the square of side length $8\ \text{cm}$ on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 8 cm by 8 cm rectangle, we want to draw as many circles of radius 2 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 8 cm wide and 8 cm tall.
Each circle has radius 2 cm, so each circle has diameter 4 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-2 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 4 cm needs a 4 cm by 4 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 2 cm has diameter 2 + 2 = 4 cm, so it fits inside a small 4 cm by 4 cm square cell.

2 + 2 = 4

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 8 cm fits 8 / 4 = 2 cells, and down 8 cm fits 8 / 4 = 2 rows of cells.

8 \div 4 = 2,\ 8 \div 4 = 2

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 2 by 2 grid of cells holds one circle per cell.

2 \times 2 = 4

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 4

4 · Reviewdoes it hold up?

4 circles is a whole number, as it must be for counting objects. The 2 columns times 2 rows of 4 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-2 circle spans a 4 cm diameter and needs a 4 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 2 x 2 = 4 total.

💡Takeaway. Each circle needs a 4 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 3 easy answer: 25

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the square of side length $10\ \text{cm}$ on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 10 cm by 10 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 10 cm wide and 10 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 10 cm fits 10 / 2 = 5 cells, and down 10 cm fits 10 / 2 = 5 rows of cells.

10 \div 2 = 5,\ 10 \div 2 = 5

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 5 by 5 grid of cells holds one circle per cell.

5 \times 5 = 25

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 25

4 · Reviewdoes it hold up?

25 circles is a whole number, as it must be for counting objects. The 5 columns times 5 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 5 x 5 = 25 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 4 medium answer: 6

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the $6\ \text{cm}$ by $4\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 6 cm by 4 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 6 cm wide and 4 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 6 cm fits 6 / 2 = 3 cells, and down 4 cm fits 4 / 2 = 2 rows of cells.

6 \div 2 = 3,\ 4 \div 2 = 2

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 3 by 2 grid of cells holds one circle per cell.

3 \times 2 = 6

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 6

4 · Reviewdoes it hold up?

6 circles is a whole number, as it must be for counting objects. The 3 columns times 2 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 3 x 2 = 6 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 5 medium answer: 12

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the $8\ \text{cm}$ by $6\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 8 cm by 6 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 8 cm wide and 6 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 8 cm fits 8 / 2 = 4 cells, and down 6 cm fits 6 / 2 = 3 rows of cells.

8 \div 2 = 4,\ 6 \div 2 = 3

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 4 by 3 grid of cells holds one circle per cell.

4 \times 3 = 12

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 12

4 · Reviewdoes it hold up?

12 circles is a whole number, as it must be for counting objects. The 4 columns times 3 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 4 x 3 = 12 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 6 medium answer: 12

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the $9\ \text{cm}$ by $6\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 9 cm by 6 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 9 cm wide and 6 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 9 cm fits 9 / 2 = 4 cells, and down 6 cm fits 6 / 2 = 3 rows of cells.

9 \div 2 = 4,\ 6 \div 2 = 3

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 4 by 3 grid of cells holds one circle per cell.

4 \times 3 = 12

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 12

4 · Reviewdoes it hold up?

12 circles is a whole number, as it must be for counting objects. The 4 columns times 3 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 4 x 3 = 12 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 7 medium answer: 10

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the $10\ \text{cm}$ by $4\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 10 cm by 4 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 10 cm wide and 4 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 10 cm fits 10 / 2 = 5 cells, and down 4 cm fits 4 / 2 = 2 rows of cells.

10 \div 2 = 5,\ 4 \div 2 = 2

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 5 by 2 grid of cells holds one circle per cell.

5 \times 2 = 10

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 10

4 · Reviewdoes it hold up?

10 circles is a whole number, as it must be for counting objects. The 5 columns times 2 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 5 x 2 = 10 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 8 hard answer: 6

You want to draw as many circles with radius $2\ \text{cm}$ as possible inside the $12\ \text{cm}$ by $8\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 12 cm by 8 cm rectangle, we want to draw as many circles of radius 2 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 12 cm wide and 8 cm tall.
Each circle has radius 2 cm, so each circle has diameter 4 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-2 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 4 cm needs a 4 cm by 4 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 2 cm has diameter 2 + 2 = 4 cm, so it fits inside a small 4 cm by 4 cm square cell.

2 + 2 = 4

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 12 cm fits 12 / 4 = 3 cells, and down 8 cm fits 8 / 4 = 2 rows of cells.

12 \div 4 = 3,\ 8 \div 4 = 2

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 3 by 2 grid of cells holds one circle per cell.

3 \times 2 = 6

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 6

4 · Reviewdoes it hold up?

6 circles is a whole number, as it must be for counting objects. The 3 columns times 2 rows of 4 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-2 circle spans a 4 cm diameter and needs a 4 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 3 x 2 = 6 total.

💡Takeaway. Each circle needs a 4 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 9 hard answer: 21

You want to draw as many circles with radius $1\ \text{cm}$ as possible inside the $14\ \text{cm}$ by $6\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 14 cm by 6 cm rectangle, we want to draw as many circles of radius 1 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 14 cm wide and 6 cm tall.
Each circle has radius 1 cm, so each circle has diameter 2 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-1 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 2 cm needs a 2 cm by 2 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 1 cm has diameter 1 + 1 = 2 cm, so it fits inside a small 2 cm by 2 cm square cell.

1 + 1 = 2

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 14 cm fits 14 / 2 = 7 cells, and down 6 cm fits 6 / 2 = 3 rows of cells.

14 \div 2 = 7,\ 6 \div 2 = 3

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 7 by 3 grid of cells holds one circle per cell.

7 \times 3 = 21

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 21

4 · Reviewdoes it hold up?

21 circles is a whole number, as it must be for counting objects. The 7 columns times 3 rows of 2 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-1 circle spans a 2 cm diameter and needs a 2 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 7 x 3 = 21 total.

💡Takeaway. Each circle needs a 2 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Variant 10 hard answer: 2

You want to draw as many circles with radius $5\ \text{cm}$ as possible inside the $20\ \text{cm}$ by $10\ \text{cm}$ rectangle on the right, without any of them overlapping. How many circles can you draw at most?

Show solution

1 · Understandwhat's really being asked

Inside a 20 cm by 10 cm rectangle, we want to draw as many circles of radius 5 cm as possible without any overlapping. We must find the largest number of circles that fit.

Givens

The rectangle is 20 cm wide and 10 cm tall.
Each circle has radius 5 cm, so each circle has diameter 10 cm.
No two circles may overlap.

Unknowns

The greatest number of non-overlapping radius-5 cm circles that fit in the rectangle.

Constraints

Each circle of diameter 10 cm needs a 10 cm by 10 cm space.
Circles must stay inside the rectangle and may touch but not overlap.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #10 Create a Physical Representation#5 Look for a Pattern

Each circle sits neatly in a square cell as wide as its diameter. Drawing the rectangle as a grid of these cells shows how many fit across and down, and the rows-times-columns pattern gives the total.

3 · Execute3 carry out the plan

1Find the space one circle needs

#1 Draw a Diagram 3.G.A.1

A circle of radius 5 cm has diameter 5 + 5 = 10 cm, so it fits inside a small 10 cm by 10 cm square cell.

5 + 5 = 10

The widest part of a circle is its diameter, which equals twice the radius.

2Count cells across and down

#5 Look for a Pattern 3.OA.C.7

Across 20 cm fits 20 / 10 = 2 cells, and down 10 cm fits 10 / 10 = 1 rows of cells.

20 \div 10 = 2,\ 10 \div 10 = 1

Splitting a length into equal pieces is division within 100.

3Multiply rows by columns

#10 Create a Physical Representation 3.OA.C.7

A 2 by 1 grid of cells holds one circle per cell.

2 \times 1 = 2

Equal rows and columns of objects are counted by multiplication, an array model from Grade 3.

Answer: 2

4 · Reviewdoes it hold up?

2 circles is a whole number, as it must be for counting objects. The 2 columns times 1 rows of 10 cm cells agree with the grid drawn in the figure.

Another way: Lay the circles out physically in a simple grid and count them directly (Tool 10, Create a Physical Representation).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Knowing each radius-5 circle spans a 10 cm diameter and needs a 10 cm cell.
3.OA.C.7 Fluently multiply and divide within 100 — Computing cells per side and 2 x 1 = 2 total.

💡Takeaway. Each circle needs a 10 cm box -- count the boxes across and down, then multiply, just like a Grade 3 array!

Rectangle sides are multiples of the diameter · 10 practice problems

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