← 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

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Generated variants — 10

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

Variant 1 easy

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.

Your answer

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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 $\frac{1}{8}$ 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

\frac{1}{8}

4 \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 \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

\frac{3}{8}

. Since 3 and 8 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{3}{8}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{5}{8}$ is unshaded and the shaded part is 1 - $\frac{5}{8}$ = $\frac{3}{8}$ .

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

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.

Your answer

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 $\frac{1}{9}$ 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 \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 \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

\frac{2}{9}

. Since 2 and 9 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{2}{9}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{7}{9}$ is unshaded and the shaded part is 1 - $\frac{7}{9}$ = $\frac{2}{9}$ .

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

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.

Your answer

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 $\frac{1}{12}$ 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

\frac{1}{12}

6 \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 \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

\frac{5}{12}

. Since 5 and 12 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{5}{12}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{7}{12}$ is unshaded and the shaded part is 1 - $\frac{7}{12}$ = $\frac{5}{12}$ .

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

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.

Your answer

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 $\frac{1}{15}$ 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

\frac{1}{15}

5 \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 \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

\frac{4}{15}

. Since 4 and 15 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{4}{15}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{11}{15}$ is unshaded and the shaded part is 1 - $\frac{11}{15}$ = $\frac{4}{15}$ .

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

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.

Your answer

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 $\frac{1}{16}$ 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

\frac{1}{16}

4 \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 \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

\frac{9}{16}

. Since 9 and 16 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{9}{16}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{7}{16}$ is unshaded and the shaded part is 1 - $\frac{7}{16}$ = $\frac{9}{16}$ .

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

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.

Your answer

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 $\frac{1}{18}$ 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

\frac{1}{18}

9 \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 \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

\frac{7}{18}

. Since 7 and 18 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{7}{18}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{11}{18}$ is unshaded and the shaded part is 1 - $\frac{11}{18}$ = $\frac{7}{18}$ .

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

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.

Your answer

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 $\frac{1}{24}$ 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

\frac{1}{24}

6 \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 \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

\frac{15}{24}

, which reduces to

\frac{5}{8}

\dfrac{5}{8}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{9}{24}$ is unshaded and the shaded part is 1 - $\frac{9}{24}$ = $\frac{15}{24}$ .

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

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.

Your answer

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 $\frac{1}{24}$ 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

\frac{1}{24}

8 \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 \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

\frac{10}{24}

, which reduces to

\frac{5}{12}

\dfrac{5}{12}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{14}{24}$ is unshaded and the shaded part is 1 - $\frac{14}{24}$ = $\frac{10}{24}$ .

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

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.

Your answer

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 $\frac{1}{32}$ 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

\frac{1}{32}

8 \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 \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

\frac{21}{32}

. Since 21 and 32 share no common factor greater than 1, the fraction is already in simplest form.

\dfrac{21}{32}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{11}{32}$ is unshaded and the shaded part is 1 - $\frac{11}{32}$ = $\frac{21}{32}$ .

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

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.

Your answer

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 $\frac{1}{50}$ 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

\frac{1}{50}

10 \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 \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

\frac{28}{50}

, which reduces to

\frac{14}{25}

\dfrac{14}{25}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer:

\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 $\frac{22}{50}$ is unshaded and the shaded part is 1 - $\frac{22}{50}$ = $\frac{28}{50}$ .

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!

Where next?

← Back to the problem: Fractions require equal partitioning of the whole