Variant 1 easy answer: 4 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $5$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$10$	$6$	$14$	$50$

(Number of students who like roses) $=$ [ ] $- (10 + 6 + 14) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $5$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (10), lilies (6), and daisies (14), with a total of 50. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 5 students.

Givens

Tulip = 10, Lily = 6, Daisy = 14 students
Total = 50 students
One vertical grid square represents 5 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 5 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

50 - (10 + 6 + 14) = 50 - 30 = 20

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 20 (rose), 10 (tulip), 6 (lily), 14 (daisy). The largest is 20, so the scale must reach at least 20 students.

\max(20, 10, 6, 14) = 20

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 5 students, so divide the largest value by 5 to find how many squares are needed.

20 \div 5 = 4

If one square holds 5 students, then 20 students fill 4 squares exactly.

Answer: 4 squares

4 · Reviewdoes it hold up?

4 squares at 5 students each reach 20 students, matching the largest value, and 20 is bigger than every other count, so 4 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 3 squares reach only 15 students, too short for 20; 4 squares reach 20, which works, confirming 4 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 5 each!

Variant 2 easy answer: 4 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $5$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$11$	$7$	$13$	$51$

(Number of students who like roses) $=$ [ ] $- (11 + 7 + 13) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $5$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (11), lilies (7), and daisies (13), with a total of 51. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 5 students.

Givens

Tulip = 11, Lily = 7, Daisy = 13 students
Total = 51 students
One vertical grid square represents 5 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 5 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

51 - (11 + 7 + 13) = 51 - 31 = 20

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 20 (rose), 11 (tulip), 7 (lily), 13 (daisy). The largest is 20, so the scale must reach at least 20 students.

\max(20, 11, 7, 13) = 20

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 5 students, so divide the largest value by 5 to find how many squares are needed.

20 \div 5 = 4

If one square holds 5 students, then 20 students fill 4 squares exactly.

Answer: 4 squares

4 · Reviewdoes it hold up?

4 squares at 5 students each reach 20 students, matching the largest value, and 20 is bigger than every other count, so 4 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 3 squares reach only 15 students, too short for 20; 4 squares reach 20, which works, confirming 4 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 5 each!

Variant 3 medium answer: 7 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $3$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$13$	$9$	$17$	$60$

(Number of students who like roses) $=$ [ ] $- (13 + 9 + 17) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $3$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (13), lilies (9), and daisies (17), with a total of 60. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 3 students.

Givens

Tulip = 13, Lily = 9, Daisy = 17 students
Total = 60 students
One vertical grid square represents 3 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 3 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

60 - (13 + 9 + 17) = 60 - 39 = 21

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 21 (rose), 13 (tulip), 9 (lily), 17 (daisy). The largest is 21, so the scale must reach at least 21 students.

\max(21, 13, 9, 17) = 21

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 3 students, so divide the largest value by 3 to find how many squares are needed.

21 \div 3 = 7

If one square holds 3 students, then 21 students fill 7 squares exactly.

Answer: 7 squares

4 · Reviewdoes it hold up?

7 squares at 3 students each reach 21 students, matching the largest value, and 21 is bigger than every other count, so 7 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 6 squares reach only 18 students, too short for 21; 7 squares reach 21, which works, confirming 7 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 3 each!

Variant 4 easy answer: 6 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $4$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$12$	$8$	$16$	$60$

(Number of students who like roses) $=$ [ ] $- (12 + 8 + 16) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $4$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (12), lilies (8), and daisies (16), with a total of 60. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 4 students.

Givens

Tulip = 12, Lily = 8, Daisy = 16 students
Total = 60 students
One vertical grid square represents 4 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 4 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

60 - (12 + 8 + 16) = 60 - 36 = 24

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 24 (rose), 12 (tulip), 8 (lily), 16 (daisy). The largest is 24, so the scale must reach at least 24 students.

\max(24, 12, 8, 16) = 24

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 4 students, so divide the largest value by 4 to find how many squares are needed.

24 \div 4 = 6

If one square holds 4 students, then 24 students fill 6 squares exactly.

Answer: 6 squares

4 · Reviewdoes it hold up?

6 squares at 4 students each reach 24 students, matching the largest value, and 24 is bigger than every other count, so 6 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 5 squares reach only 20 students, too short for 24; 6 squares reach 24, which works, confirming 6 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 4 each!

Variant 5 medium answer: 11 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $2$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$18$	$10$	$14$	$64$

(Number of students who like roses) $=$ [ ] $- (18 + 10 + 14) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $2$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (18), lilies (10), and daisies (14), with a total of 64. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 2 students.

Givens

Tulip = 18, Lily = 10, Daisy = 14 students
Total = 64 students
One vertical grid square represents 2 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 2 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

64 - (18 + 10 + 14) = 64 - 42 = 22

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 22 (rose), 18 (tulip), 10 (lily), 14 (daisy). The largest is 22, so the scale must reach at least 22 students.

\max(22, 18, 10, 14) = 22

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 2 students, so divide the largest value by 2 to find how many squares are needed.

22 \div 2 = 11

If one square holds 2 students, then 22 students fill 11 squares exactly.

Answer: 11 squares

4 · Reviewdoes it hold up?

11 squares at 2 students each reach 22 students, matching the largest value, and 22 is bigger than every other count, so 11 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 10 squares reach only 20 students, too short for 22; 11 squares reach 22, which works, confirming 11 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 2 each!

Variant 6 medium answer: 5 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $5$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$15$	$9$	$21$	$70$

(Number of students who like roses) $=$ [ ] $- (15 + 9 + 21) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $5$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (15), lilies (9), and daisies (21), with a total of 70. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 5 students.

Givens

Tulip = 15, Lily = 9, Daisy = 21 students
Total = 70 students
One vertical grid square represents 5 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 5 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

70 - (15 + 9 + 21) = 70 - 45 = 25

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 25 (rose), 15 (tulip), 9 (lily), 21 (daisy). The largest is 25, so the scale must reach at least 25 students.

\max(25, 15, 9, 21) = 25

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 5 students, so divide the largest value by 5 to find how many squares are needed.

25 \div 5 = 5

If one square holds 5 students, then 25 students fill 5 squares exactly.

Answer: 5 squares

4 · Reviewdoes it hold up?

5 squares at 5 students each reach 25 students, matching the largest value, and 25 is bigger than every other count, so 5 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 4 squares reach only 20 students, too short for 25; 5 squares reach 25, which works, confirming 5 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 5 each!

Variant 7 medium answer: 15 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $2$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$16$	$12$	$20$	$78$

(Number of students who like roses) $=$ [ ] $- (16 + 12 + 20) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $2$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (16), lilies (12), and daisies (20), with a total of 78. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 2 students.

Givens

Tulip = 16, Lily = 12, Daisy = 20 students
Total = 78 students
One vertical grid square represents 2 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 2 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

78 - (16 + 12 + 20) = 78 - 48 = 30

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 30 (rose), 16 (tulip), 12 (lily), 20 (daisy). The largest is 30, so the scale must reach at least 30 students.

\max(30, 16, 12, 20) = 30

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 2 students, so divide the largest value by 2 to find how many squares are needed.

30 \div 2 = 15

If one square holds 2 students, then 30 students fill 15 squares exactly.

Answer: 15 squares

4 · Reviewdoes it hold up?

15 squares at 2 students each reach 30 students, matching the largest value, and 30 is bigger than every other count, so 15 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 14 squares reach only 28 students, too short for 30; 15 squares reach 30, which works, confirming 15 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 2 each!

Variant 8 hard answer: 14 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $2$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$20$	$14$	$18$	$80$

(Number of students who like roses) $=$ [ ] $- (20 + 14 + 18) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $2$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (20), lilies (14), and daisies (18), with a total of 80. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 2 students.

Givens

Tulip = 20, Lily = 14, Daisy = 18 students
Total = 80 students
One vertical grid square represents 2 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 2 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

80 - (20 + 14 + 18) = 80 - 52 = 28

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 28 (rose), 20 (tulip), 14 (lily), 18 (daisy). The largest is 28, so the scale must reach at least 28 students.

\max(28, 20, 14, 18) = 28

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 2 students, so divide the largest value by 2 to find how many squares are needed.

28 \div 2 = 14

If one square holds 2 students, then 28 students fill 14 squares exactly.

Answer: 14 squares

4 · Reviewdoes it hold up?

14 squares at 2 students each reach 28 students, matching the largest value, and 28 is bigger than every other count, so 14 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 13 squares reach only 26 students, too short for 28; 14 squares reach 28, which works, confirming 14 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 2 each!

Variant 9 hard answer: 17 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $2$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$22$	$18$	$26$	$100$

(Number of students who like roses) $=$ [ ] $- (22 + 18 + 26) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $2$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (22), lilies (18), and daisies (26), with a total of 100. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 2 students.

Givens

Tulip = 22, Lily = 18, Daisy = 26 students
Total = 100 students
One vertical grid square represents 2 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 2 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

100 - (22 + 18 + 26) = 100 - 66 = 34

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 34 (rose), 22 (tulip), 18 (lily), 26 (daisy). The largest is 34, so the scale must reach at least 34 students.

\max(34, 22, 18, 26) = 34

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 2 students, so divide the largest value by 2 to find how many squares are needed.

34 \div 2 = 17

If one square holds 2 students, then 34 students fill 17 squares exactly.

Answer: 17 squares

4 · Reviewdoes it hold up?

17 squares at 2 students each reach 34 students, matching the largest value, and 34 is bigger than every other count, so 17 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 16 squares reach only 32 students, too short for 34; 17 squares reach 34, which works, confirming 17 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 2 each!

Variant 10 hard answer: 9 squares

Representative Problem

The table shows the favorite flowers of the students at Mia's school. If this table is drawn as a bar graph in which one vertical grid square represents $4$ students, find the least number of vertical grid squares the scale must have.

Table "Favorite Flower":

Flower	Rose	Tulip	Lily	Daisy	Total
Number of students	(?)	$24$	$16$	$28$	$104$

(Number of students who like roses) $=$ [ ] $- (24 + 16 + 28) =$ [ ] (students)
Since [ ] students like roses, the vertical scale must reach at least [ ] students.
If one vertical grid square represents $4$ students, the vertical scale needs at least [ ] squares.

Show solution

1 · Understandwhat's really being asked

A table lists how many students like roses, tulips (24), lilies (16), and daisies (28), with a total of 104. First find the missing rose count, then decide how many vertical grid squares a bar graph needs if one square stands for 4 students.

Givens

Tulip = 24, Lily = 16, Daisy = 28 students
Total = 104 students
One vertical grid square represents 4 students

Unknowns

The number of students who like roses
The least number of vertical grid squares the scale must have

Constraints

The scale must reach at least the largest single value
Each grid square stands for exactly 4 students

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #8 Analyze the Units

This is a two-step task: first recover the missing value from the total (a subtraction subproblem), then convert the largest value into grid squares using the 'students per square' unit.

3 · Execute3 carry out the plan

1Find the rose count

#7 Identify Subproblems 3.MD.B.3

Subtract the three known categories from the total to get the number of students who like roses.

104 - (24 + 16 + 28) = 104 - 68 = 36

A total is the sum of its parts, so the missing part is the total minus the known parts.

2Find the largest value

#7 Identify Subproblems 3.MD.B.3

Compare the four counts: 36 (rose), 24 (tulip), 16 (lily), 28 (daisy). The largest is 36, so the scale must reach at least 36 students.

\max(36, 24, 16, 28) = 36

The tallest bar sets how high the scale must go.

3Convert students to grid squares

#8 Analyze the Units 3.MD.B.3

Each grid square is worth 4 students, so divide the largest value by 4 to find how many squares are needed.

36 \div 4 = 9

If one square holds 4 students, then 36 students fill 9 squares exactly.

Answer: 9 squares

4 · Reviewdoes it hold up?

9 squares at 4 students each reach 36 students, matching the largest value, and 36 is bigger than every other count, so 9 squares is the smallest scale that fits.

Another way: Guess and check (tool 6): 8 squares reach only 32 students, too short for 36; 9 squares reach 36, which works, confirming 9 is the least.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Choosing a vertical scale that fits the largest data value and counting the needed grid squares

💡Takeaway. This only needs Grade 3 bar-graph sense: find the biggest value, then count squares worth 4 each!

Choose a scale that fits the largest value · 10 practice problems

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