← 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: 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 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 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 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: 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 5/8 is unshaded and the shaded part is 1 - 5/8 = 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 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: 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 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 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: 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 7/9 is unshaded and the shaded part is 1 - 7/9 = 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 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: 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 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 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 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: 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 7/12 is unshaded and the shaded part is 1 - 7/12 = 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 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: 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 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 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 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: 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 11/15 is unshaded and the shaded part is 1 - 11/15 = 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 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: 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 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 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 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: 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 7/16 is unshaded and the shaded part is 1 - 7/16 = 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 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: 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 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 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 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: 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 11/18 is unshaded and the shaded part is 1 - 11/18 = 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 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: 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 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 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 15/24, which reduces to 5/8.
58\dfrac{5}{8}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 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 9/24 is unshaded and the shaded part is 1 - 9/24 = 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 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: 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 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 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 10/24, which reduces to 5/12.
512\dfrac{5}{12}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 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 14/24 is unshaded and the shaded part is 1 - 14/24 = 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 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: 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 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 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 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: 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 11/32 is unshaded and the shaded part is 1 - 11/32 = 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 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: 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 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 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 28/50, which reduces to 14/25.
1425\dfrac{14}{25}
The shaded part as a fraction of the whole is just shaded cells over all cells.
Answer: 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 22/50 is unshaded and the shaded part is 1 - 22/50 = 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 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!