← Fractions require equal partitioning of the whole · Part-Whole Fraction Reasoning

Fractions require equal partitioning of the whole · 10 practice problems

3.NF.A.13.G.A.2

Generated variants — 10

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

Variant 1 easy answer: $\frac{3}{8}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 8 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 8 equal cells (a 4-by-2 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 8 equal-size cells (4 columns by 2 rows)
  • Each cell is the same size, so each cell is 18\frac{1}{8} of the whole
  • From the figure, the shaded region covers 3 of the 4 columns across the top 1 of the 2 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 4-by-2 array of equal cells, so the whole is 8 equal parts and each cell is 18\frac{1}{8}.
4×2=84 \times 2 = 8
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 3 of the 4 columns and the top 1 of the 2 rows, so it covers 3 times 1 = 3 cells out of 8.
3×1=3 shaded cells out of 83 \times 1 = 3 \text{ shaded cells out of } 8
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 38\frac{3}{8}. Since 3 and 8 share no common factor greater than 1, the fraction is already in simplest form.
38\dfrac{3}{8}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 38\frac{3}{8}
4 · Reviewdoes it hold up?

The unshaded region is 5 of the 8 cells, so the shaded part is 3 of 8, matching the picture.

Another way: Change focus to the complement (tool 16): 5 cells are unshaded, so 58\frac{5}{8} is unshaded and the shaded part is 1 - 58\frac{5}{8} = 38\frac{3}{8}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{3}{8}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 8 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 2 easy answer: $\frac{2}{9}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 9 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 9 equal cells (a 3-by-3 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 9 equal-size cells (3 columns by 3 rows)
  • Each cell is the same size, so each cell is 19\frac{1}{9} of the whole
  • From the figure, the shaded region covers 2 of the 3 columns across the top 1 of the 3 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 3-by-3 array of equal cells, so the whole is 9 equal parts and each cell is 1/9.
3×3=93 \times 3 = 9
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 2 of the 3 columns and the top 1 of the 3 rows, so it covers 2 times 1 = 2 cells out of 9.
2×1=2 shaded cells out of 92 \times 1 = 2 \text{ shaded cells out of } 9
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 29\frac{2}{9}. Since 2 and 9 share no common factor greater than 1, the fraction is already in simplest form.
29\dfrac{2}{9}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 29\frac{2}{9}
4 · Reviewdoes it hold up?

The unshaded region is 7 of the 9 cells, so the shaded part is 2 of 9, matching the picture.

Another way: Change focus to the complement (tool 16): 7 cells are unshaded, so 79\frac{7}{9} is unshaded and the shaded part is 1 - 79\frac{7}{9} = 29\frac{2}{9}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{2}{9}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 9 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 3 easy answer: $\frac{5}{12}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 12 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 12 equal cells (a 6-by-2 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 12 equal-size cells (6 columns by 2 rows)
  • Each cell is the same size, so each cell is 112\frac{1}{12} of the whole
  • From the figure, the shaded region covers 5 of the 6 columns across the top 1 of the 2 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 6-by-2 array of equal cells, so the whole is 12 equal parts and each cell is 112\frac{1}{12}.
6×2=126 \times 2 = 12
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 5 of the 6 columns and the top 1 of the 2 rows, so it covers 5 times 1 = 5 cells out of 12.
5×1=5 shaded cells out of 125 \times 1 = 5 \text{ shaded cells out of } 12
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 512\frac{5}{12}. Since 5 and 12 share no common factor greater than 1, the fraction is already in simplest form.
512\dfrac{5}{12}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 512\frac{5}{12}
4 · Reviewdoes it hold up?

The unshaded region is 7 of the 12 cells, so the shaded part is 5 of 12, matching the picture.

Another way: Change focus to the complement (tool 16): 7 cells are unshaded, so 712\frac{7}{12} is unshaded and the shaded part is 1 - 712\frac{7}{12} = 512\frac{5}{12}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{5}{12}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 12 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 4 medium answer: $\frac{4}{15}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 15 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 15 equal cells (a 5-by-3 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 15 equal-size cells (5 columns by 3 rows)
  • Each cell is the same size, so each cell is 115\frac{1}{15} of the whole
  • From the figure, the shaded region covers 2 of the 5 columns across the top 2 of the 3 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 5-by-3 array of equal cells, so the whole is 15 equal parts and each cell is 115\frac{1}{15}.
5×3=155 \times 3 = 15
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 2 of the 5 columns and the top 2 of the 3 rows, so it covers 2 times 2 = 4 cells out of 15.
2×2=4 shaded cells out of 152 \times 2 = 4 \text{ shaded cells out of } 15
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 415\frac{4}{15}. Since 4 and 15 share no common factor greater than 1, the fraction is already in simplest form.
415\dfrac{4}{15}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 415\frac{4}{15}
4 · Reviewdoes it hold up?

The unshaded region is 11 of the 15 cells, so the shaded part is 4 of 15, matching the picture.

Another way: Change focus to the complement (tool 16): 11 cells are unshaded, so 1115\frac{11}{15} is unshaded and the shaded part is 1 - 1115\frac{11}{15} = 415\frac{4}{15}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{4}{15}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 15 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 5 medium answer: $\frac{9}{16}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 16 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 16 equal cells (a 4-by-4 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 16 equal-size cells (4 columns by 4 rows)
  • Each cell is the same size, so each cell is 116\frac{1}{16} of the whole
  • From the figure, the shaded region covers 3 of the 4 columns across the top 3 of the 4 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 4-by-4 array of equal cells, so the whole is 16 equal parts and each cell is 116\frac{1}{16}.
4×4=164 \times 4 = 16
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 3 of the 4 columns and the top 3 of the 4 rows, so it covers 3 times 3 = 9 cells out of 16.
3×3=9 shaded cells out of 163 \times 3 = 9 \text{ shaded cells out of } 16
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 916\frac{9}{16}. Since 9 and 16 share no common factor greater than 1, the fraction is already in simplest form.
916\dfrac{9}{16}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 916\frac{9}{16}
4 · Reviewdoes it hold up?

The unshaded region is 7 of the 16 cells, so the shaded part is 9 of 16, matching the picture.

Another way: Change focus to the complement (tool 16): 7 cells are unshaded, so 716\frac{7}{16} is unshaded and the shaded part is 1 - 716\frac{7}{16} = 916\frac{9}{16}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{9}{16}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 16 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 6 medium answer: $\frac{7}{18}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 18 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 18 equal cells (a 9-by-2 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 18 equal-size cells (9 columns by 2 rows)
  • Each cell is the same size, so each cell is 118\frac{1}{18} of the whole
  • From the figure, the shaded region covers 7 of the 9 columns across the top 1 of the 2 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 9-by-2 array of equal cells, so the whole is 18 equal parts and each cell is 118\frac{1}{18}.
9×2=189 \times 2 = 18
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 7 of the 9 columns and the top 1 of the 2 rows, so it covers 7 times 1 = 7 cells out of 18.
7×1=7 shaded cells out of 187 \times 1 = 7 \text{ shaded cells out of } 18
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 718\frac{7}{18}. Since 7 and 18 share no common factor greater than 1, the fraction is already in simplest form.
718\dfrac{7}{18}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 718\frac{7}{18}
4 · Reviewdoes it hold up?

The unshaded region is 11 of the 18 cells, so the shaded part is 7 of 18, matching the picture.

Another way: Change focus to the complement (tool 16): 11 cells are unshaded, so 1118\frac{11}{18} is unshaded and the shaded part is 1 - 1118\frac{11}{18} = 718\frac{7}{18}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{7}{18}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 18 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 7 medium answer: $\frac{5}{8}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 24 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 24 equal cells (a 6-by-4 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 24 equal-size cells (6 columns by 4 rows)
  • Each cell is the same size, so each cell is 124\frac{1}{24} of the whole
  • From the figure, the shaded region covers 5 of the 6 columns across the top 3 of the 4 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 6-by-4 array of equal cells, so the whole is 24 equal parts and each cell is 124\frac{1}{24}.
6×4=246 \times 4 = 24
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 5 of the 6 columns and the top 3 of the 4 rows, so it covers 5 times 3 = 15 cells out of 24.
5×3=15 shaded cells out of 245 \times 3 = 15 \text{ shaded cells out of } 24
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 1524\frac{15}{24}, which reduces to 58\frac{5}{8}.
58\dfrac{5}{8}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 58\frac{5}{8}
4 · Reviewdoes it hold up?

The unshaded region is 9 of the 24 cells, so the shaded part is 15 of 24, matching the picture.

Another way: Change focus to the complement (tool 16): 9 cells are unshaded, so 924\frac{9}{24} is unshaded and the shaded part is 1 - 924\frac{9}{24} = 1524\frac{15}{24}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{5}{8}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 24 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 8 hard answer: $\frac{5}{12}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 24 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 24 equal cells (a 8-by-3 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 24 equal-size cells (8 columns by 3 rows)
  • Each cell is the same size, so each cell is 124\frac{1}{24} of the whole
  • From the figure, the shaded region covers 5 of the 8 columns across the top 2 of the 3 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 8-by-3 array of equal cells, so the whole is 24 equal parts and each cell is 124\frac{1}{24}.
8×3=248 \times 3 = 24
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 5 of the 8 columns and the top 2 of the 3 rows, so it covers 5 times 2 = 10 cells out of 24.
5×2=10 shaded cells out of 245 \times 2 = 10 \text{ shaded cells out of } 24
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 1024\frac{10}{24}, which reduces to 512\frac{5}{12}.
512\dfrac{5}{12}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 512\frac{5}{12}
4 · Reviewdoes it hold up?

The unshaded region is 14 of the 24 cells, so the shaded part is 10 of 24, matching the picture.

Another way: Change focus to the complement (tool 16): 14 cells are unshaded, so 1424\frac{14}{24} is unshaded and the shaded part is 1 - 1424\frac{14}{24} = 1024\frac{10}{24}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{5}{12}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 24 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 9 hard answer: $\frac{21}{32}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 32 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 32 equal cells (a 8-by-4 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 32 equal-size cells (8 columns by 4 rows)
  • Each cell is the same size, so each cell is 132\frac{1}{32} of the whole
  • From the figure, the shaded region covers 7 of the 8 columns across the top 3 of the 4 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 8-by-4 array of equal cells, so the whole is 32 equal parts and each cell is 132\frac{1}{32}.
8×4=328 \times 4 = 32
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 7 of the 8 columns and the top 3 of the 4 rows, so it covers 7 times 3 = 21 cells out of 32.
7×3=21 shaded cells out of 327 \times 3 = 21 \text{ shaded cells out of } 32
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 2132\frac{21}{32}. Since 21 and 32 share no common factor greater than 1, the fraction is already in simplest form.
2132\dfrac{21}{32}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 2132\frac{21}{32}
4 · Reviewdoes it hold up?

The unshaded region is 11 of the 32 cells, so the shaded part is 21 of 32, matching the picture.

Another way: Change focus to the complement (tool 16): 11 cells are unshaded, so 1132\frac{11}{32} is unshaded and the shaded part is 1 - 1132\frac{11}{32} = 2132\frac{21}{32}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{21}{32}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 32 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!
Variant 10 hard answer: $\frac{14}{25}$

Write a fraction that tells what part of the whole is shaded.

(Figure) A rectangle has tick marks along each side that divide the whole rectangle into 50 equal-size cells. A portion of the rectangle is shaded.

Show solution
1 · Understandwhat's really being asked

A rectangle is partitioned by tick marks into 50 equal cells (a 10-by-5 grid). Part of it is shaded. Write, as a fraction, what part of the whole rectangle is shaded.

Givens
  • The whole rectangle is divided into 50 equal-size cells (10 columns by 5 rows)
  • Each cell is the same size, so each cell is 150\frac{1}{50} of the whole
  • From the figure, the shaded region covers 7 of the 10 columns across the top 4 of the 5 rows
Unknowns
  • The fraction of the whole rectangle that is shaded
Constraints
  • The whole must be divided into equal parts to name a fraction
  • The fraction is shaded cells over total cells, written in simplest form
2 · Planchoose the strategy

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

The grid is already a diagram of equal parts, so we count shaded cells against the total. Breaking the count into rows and columns (a subproblem) makes the shaded count quick and reliable.

3 · Execute3 carry out the plan

1Confirm equal parts

#1 Draw a Diagram 3.G.A.2
The tick marks split the rectangle into a 10-by-5 array of equal cells, so the whole is 50 equal parts and each cell is 150\frac{1}{50}.
10×5=5010 \times 5 = 50
Equal partitioning is what lets us name the shaded amount with a single fraction.

2Count the shaded cells

#7 Identify Subproblems 3.NF.A.1
The shaded region spans 7 of the 10 columns and the top 4 of the 5 rows, so it covers 7 times 4 = 28 cells out of 50.
7×4=28 shaded cells out of 507 \times 4 = 28 \text{ shaded cells out of } 50
Counting full rows of equal cells is faster than counting one cell at a time.

3Write the fraction

#1 Draw a Diagram 3.NF.A.1
Shaded over total is 2850\frac{28}{50}, which reduces to 1425\frac{14}{25}.
1425\dfrac{14}{25}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 1425\frac{14}{25}
4 · Reviewdoes it hold up?

The unshaded region is 22 of the 50 cells, so the shaded part is 28 of 50, matching the picture.

Another way: Change focus to the complement (tool 16): 22 cells are unshaded, so 2250\frac{22}{50} is unshaded and the shaded part is 1 - 2250\frac{22}{50} = 2850\frac{28}{50}.

Standardsmin grade 3
  • 3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Naming the shaded amount as $\frac{14}{25}$ of the whole
  • 3.G.A.2 Partition shapes into equal parts with equal areas — Recognizing the 50 cells as equal parts of the rectangle
💡Takeaway. When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!