Practice · Set the scale, then complete the graph · Sensim Math

Variant 1 easy

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $8$ students.
The number of students in Class 2 is $2$ fewer than the number in Class 3.
The four classes have $30$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 8 students, Class 2 is 2 fewer than Class 3, and all four classes total 30), I set the value of one grid square and fill in each class.

Givens

Class 1 has 8 students.
Class 2 has 2 fewer students than Class 3.
All four classes together have 30 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 30.

2 · Planchoose the strategy

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

The 2-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 2 students). With the scale known I work out each class in small steps and use the total of 30 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 2 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 2 / 1 = 2 students.

2 \text{ students} \div 1 \text{ circle} = 2 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 2 students per circle: Class 1 = 8 / 2 = 4 circles, and Class 3 = 8 students = 4 circles. Class 2 is one circle fewer: 3 circles = 6 students.

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

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 30. Subtract the three known classes: 30 - 8 - 6 - 8 = 8, so Class 4 has 8 students.

30 - 8 - 6 - 8 = 8

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 2 students per circle: Class 1 = 4, Class 2 = 3, Class 3 = 4, Class 4 = 8 / 2 = 4. Draw circles from the bottom up to those heights.

8 \div 2 = 4

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 2 students; Class 1 = 8 (4 circles), Class 2 = 6 (3 circles), Class 3 = 8 (4 circles), Class 4 = 8 (4 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 8 + 6 + 8 + 8 = 30, matching the total, and Class 2 (6) is exactly 2 fewer than Class 3 (8). Both conditions hold.

Another way: Knowing Class 1 = 8 (4 circles) sets the scale at 8 / 4 = 2 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 2 easy

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $9$ students.
The number of students in Class 2 is $3$ fewer than the number in Class 3.
The four classes have $33$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 9 students, Class 2 is 3 fewer than Class 3, and all four classes total 33), I set the value of one grid square and fill in each class.

Givens

Class 1 has 9 students.
Class 2 has 3 fewer students than Class 3.
All four classes together have 33 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 33.

2 · Planchoose the strategy

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

The 3-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 3 students). With the scale known I work out each class in small steps and use the total of 33 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 3 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 3 / 1 = 3 students.

3 \text{ students} \div 1 \text{ circle} = 3 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 3 students per circle: Class 1 = 9 / 3 = 3 circles, and Class 3 = 9 students = 3 circles. Class 2 is one circle fewer: 2 circles = 6 students.

9 \div 3 = 3,\quad 9 \div 3 = 3

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 33. Subtract the three known classes: 33 - 9 - 6 - 9 = 9, so Class 4 has 9 students.

33 - 9 - 6 - 9 = 9

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 3 students per circle: Class 1 = 3, Class 2 = 2, Class 3 = 3, Class 4 = 9 / 3 = 3. Draw circles from the bottom up to those heights.

9 \div 3 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 3 students; Class 1 = 9 (3 circles), Class 2 = 6 (2 circles), Class 3 = 9 (3 circles), Class 4 = 9 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 9 + 6 + 9 + 9 = 33, matching the total, and Class 2 (6) is exactly 3 fewer than Class 3 (9). Both conditions hold.

Another way: Knowing Class 1 = 9 (3 circles) sets the scale at 9 / 3 = 3 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 3 easy

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $10$ students.
The number of students in Class 2 is $2$ fewer than the number in Class 3.
The four classes have $34$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 10 students, Class 2 is 2 fewer than Class 3, and all four classes total 34), I set the value of one grid square and fill in each class.

Givens

Class 1 has 10 students.
Class 2 has 2 fewer students than Class 3.
All four classes together have 34 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 34.

2 · Planchoose the strategy

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

The 2-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 2 students). With the scale known I work out each class in small steps and use the total of 34 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 2 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 2 / 1 = 2 students.

2 \text{ students} \div 1 \text{ circle} = 2 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 2 students per circle: Class 1 = 10 / 2 = 5 circles, and Class 3 = 10 students = 5 circles. Class 2 is one circle fewer: 4 circles = 8 students.

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

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 34. Subtract the three known classes: 34 - 10 - 8 - 10 = 6, so Class 4 has 6 students.

34 - 10 - 8 - 10 = 6

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 2 students per circle: Class 1 = 5, Class 2 = 4, Class 3 = 5, Class 4 = 6 / 2 = 3. Draw circles from the bottom up to those heights.

6 \div 2 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 2 students; Class 1 = 10 (5 circles), Class 2 = 8 (4 circles), Class 3 = 10 (5 circles), Class 4 = 6 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 10 + 8 + 10 + 6 = 34, matching the total, and Class 2 (8) is exactly 2 fewer than Class 3 (10). Both conditions hold.

Another way: Knowing Class 1 = 10 (5 circles) sets the scale at 10 / 5 = 2 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 4 medium

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $12$ students.
The number of students in Class 2 is $3$ fewer than the number in Class 3.
The four classes have $42$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 12 students, Class 2 is 3 fewer than Class 3, and all four classes total 42), I set the value of one grid square and fill in each class.

Givens

Class 1 has 12 students.
Class 2 has 3 fewer students than Class 3.
All four classes together have 42 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 42.

2 · Planchoose the strategy

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

The 3-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 3 students). With the scale known I work out each class in small steps and use the total of 42 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 3 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 3 / 1 = 3 students.

3 \text{ students} \div 1 \text{ circle} = 3 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 3 students per circle: Class 1 = 12 / 3 = 4 circles, and Class 3 = 12 students = 4 circles. Class 2 is one circle fewer: 3 circles = 9 students.

12 \div 3 = 4,\quad 12 \div 3 = 4

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 42. Subtract the three known classes: 42 - 12 - 9 - 12 = 9, so Class 4 has 9 students.

42 - 12 - 9 - 12 = 9

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 3 students per circle: Class 1 = 4, Class 2 = 3, Class 3 = 4, Class 4 = 9 / 3 = 3. Draw circles from the bottom up to those heights.

9 \div 3 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 3 students; Class 1 = 12 (4 circles), Class 2 = 9 (3 circles), Class 3 = 12 (4 circles), Class 4 = 9 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 12 + 9 + 12 + 9 = 42, matching the total, and Class 2 (9) is exactly 3 fewer than Class 3 (12). Both conditions hold.

Another way: Knowing Class 1 = 12 (4 circles) sets the scale at 12 / 4 = 3 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 5 medium

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $12$ students.
The number of students in Class 2 is $4$ fewer than the number in Class 3.
The four classes have $44$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 12 students, Class 2 is 4 fewer than Class 3, and all four classes total 44), I set the value of one grid square and fill in each class.

Givens

Class 1 has 12 students.
Class 2 has 4 fewer students than Class 3.
All four classes together have 44 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 44.

2 · Planchoose the strategy

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

The 4-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 4 students). With the scale known I work out each class in small steps and use the total of 44 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 4 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 4 / 1 = 4 students.

4 \text{ students} \div 1 \text{ circle} = 4 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 4 students per circle: Class 1 = 12 / 4 = 3 circles, and Class 3 = 12 students = 3 circles. Class 2 is one circle fewer: 2 circles = 8 students.

12 \div 4 = 3,\quad 12 \div 4 = 3

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 44. Subtract the three known classes: 44 - 12 - 8 - 12 = 12, so Class 4 has 12 students.

44 - 12 - 8 - 12 = 12

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 4 students per circle: Class 1 = 3, Class 2 = 2, Class 3 = 3, Class 4 = 12 / 4 = 3. Draw circles from the bottom up to those heights.

12 \div 4 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 4 students; Class 1 = 12 (3 circles), Class 2 = 8 (2 circles), Class 3 = 12 (3 circles), Class 4 = 12 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 12 + 8 + 12 + 12 = 44, matching the total, and Class 2 (8) is exactly 4 fewer than Class 3 (12). Both conditions hold.

Another way: Knowing Class 1 = 12 (3 circles) sets the scale at 12 / 3 = 4 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 6 medium

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $15$ students.
The number of students in Class 2 is $5$ fewer than the number in Class 3.
The four classes have $50$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 15 students, Class 2 is 5 fewer than Class 3, and all four classes total 50), I set the value of one grid square and fill in each class.

Givens

Class 1 has 15 students.
Class 2 has 5 fewer students than Class 3.
All four classes together have 50 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 50.

2 · Planchoose the strategy

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

The 5-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 5 students). With the scale known I work out each class in small steps and use the total of 50 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 5 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 5 / 1 = 5 students.

5 \text{ students} \div 1 \text{ circle} = 5 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 5 students per circle: Class 1 = 15 / 5 = 3 circles, and Class 3 = 15 students = 3 circles. Class 2 is one circle fewer: 2 circles = 10 students.

15 \div 5 = 3,\quad 15 \div 5 = 3

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 50. Subtract the three known classes: 50 - 15 - 10 - 15 = 10, so Class 4 has 10 students.

50 - 15 - 10 - 15 = 10

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 5 students per circle: Class 1 = 3, Class 2 = 2, Class 3 = 3, Class 4 = 10 / 5 = 2. Draw circles from the bottom up to those heights.

10 \div 5 = 2

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 5 students; Class 1 = 15 (3 circles), Class 2 = 10 (2 circles), Class 3 = 15 (3 circles), Class 4 = 10 (2 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 15 + 10 + 15 + 10 = 50, matching the total, and Class 2 (10) is exactly 5 fewer than Class 3 (15). Both conditions hold.

Another way: Knowing Class 1 = 15 (3 circles) sets the scale at 15 / 3 = 5 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 7 medium

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $14$ students.
The number of students in Class 2 is $2$ fewer than the number in Class 3.
The four classes have $50$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 14 students, Class 2 is 2 fewer than Class 3, and all four classes total 50), I set the value of one grid square and fill in each class.

Givens

Class 1 has 14 students.
Class 2 has 2 fewer students than Class 3.
All four classes together have 50 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 50.

2 · Planchoose the strategy

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

The 2-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 2 students). With the scale known I work out each class in small steps and use the total of 50 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 2 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 2 / 1 = 2 students.

2 \text{ students} \div 1 \text{ circle} = 2 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 2 students per circle: Class 1 = 14 / 2 = 7 circles, and Class 3 = 14 students = 7 circles. Class 2 is one circle fewer: 6 circles = 12 students.

14 \div 2 = 7,\quad 14 \div 2 = 7

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 50. Subtract the three known classes: 50 - 14 - 12 - 14 = 10, so Class 4 has 10 students.

50 - 14 - 12 - 14 = 10

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 2 students per circle: Class 1 = 7, Class 2 = 6, Class 3 = 7, Class 4 = 10 / 2 = 5. Draw circles from the bottom up to those heights.

10 \div 2 = 5

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 2 students; Class 1 = 14 (7 circles), Class 2 = 12 (6 circles), Class 3 = 14 (7 circles), Class 4 = 10 (5 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 14 + 12 + 14 + 10 = 50, matching the total, and Class 2 (12) is exactly 2 fewer than Class 3 (14). Both conditions hold.

Another way: Knowing Class 1 = 14 (7 circles) sets the scale at 14 / 7 = 2 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 8 hard

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $16$ students.
The number of students in Class 2 is $4$ fewer than the number in Class 3.
The four classes have $56$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 16 students, Class 2 is 4 fewer than Class 3, and all four classes total 56), I set the value of one grid square and fill in each class.

Givens

Class 1 has 16 students.
Class 2 has 4 fewer students than Class 3.
All four classes together have 56 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 56.

2 · Planchoose the strategy

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

The 4-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 4 students). With the scale known I work out each class in small steps and use the total of 56 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 4 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 4 / 1 = 4 students.

4 \text{ students} \div 1 \text{ circle} = 4 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 4 students per circle: Class 1 = 16 / 4 = 4 circles, and Class 3 = 16 students = 4 circles. Class 2 is one circle fewer: 3 circles = 12 students.

16 \div 4 = 4,\quad 16 \div 4 = 4

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 56. Subtract the three known classes: 56 - 16 - 12 - 16 = 12, so Class 4 has 12 students.

56 - 16 - 12 - 16 = 12

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 4 students per circle: Class 1 = 4, Class 2 = 3, Class 3 = 4, Class 4 = 12 / 4 = 3. Draw circles from the bottom up to those heights.

12 \div 4 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 4 students; Class 1 = 16 (4 circles), Class 2 = 12 (3 circles), Class 3 = 16 (4 circles), Class 4 = 12 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 16 + 12 + 16 + 12 = 56, matching the total, and Class 2 (12) is exactly 4 fewer than Class 3 (16). Both conditions hold.

Another way: Knowing Class 1 = 16 (4 circles) sets the scale at 16 / 4 = 4 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 9 hard

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $18$ students.
The number of students in Class 2 is $3$ fewer than the number in Class 3.
The four classes have $60$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 18 students, Class 2 is 3 fewer than Class 3, and all four classes total 60), I set the value of one grid square and fill in each class.

Givens

Class 1 has 18 students.
Class 2 has 3 fewer students than Class 3.
All four classes together have 60 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 60.

2 · Planchoose the strategy

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

The 3-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 3 students). With the scale known I work out each class in small steps and use the total of 60 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 3 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 3 / 1 = 3 students.

3 \text{ students} \div 1 \text{ circle} = 3 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 3 students per circle: Class 1 = 18 / 3 = 6 circles, and Class 3 = 18 students = 6 circles. Class 2 is one circle fewer: 5 circles = 15 students.

18 \div 3 = 6,\quad 18 \div 3 = 6

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 60. Subtract the three known classes: 60 - 18 - 15 - 18 = 9, so Class 4 has 9 students.

60 - 18 - 15 - 18 = 9

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 3 students per circle: Class 1 = 6, Class 2 = 5, Class 3 = 6, Class 4 = 9 / 3 = 3. Draw circles from the bottom up to those heights.

9 \div 3 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 3 students; Class 1 = 18 (6 circles), Class 2 = 15 (5 circles), Class 3 = 18 (6 circles), Class 4 = 9 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 18 + 15 + 18 + 9 = 60, matching the total, and Class 2 (15) is exactly 3 fewer than Class 3 (18). Both conditions hold.

Another way: Knowing Class 1 = 18 (6 circles) sets the scale at 18 / 6 = 3 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Variant 10 hard

Mr. Carter surveyed the number of students in each second-grade class at his school and wants to show the results on a graph. Complete the graph so that it matches the conditions below.

Class 1 has $20$ students.
The number of students in Class 2 is $4$ fewer than the number in Class 3.
The four classes have $68$ students in all.

The graph is a "Students per Class" picture graph drawn with circles (o). The horizontal axis lists Class 1, Class 2, Class 3, and Class 4, and the vertical axis shows "Number of students." First decide how many students one vertical grid square should stand for, then draw a circle (o) for each class from the bottom up to complete the graph.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

I must complete a 'Students per Class' picture graph for four classes. Using the conditions (Class 1 = 20 students, Class 2 is 4 fewer than Class 3, and all four classes total 68), I set the value of one grid square and fill in each class.

Givens

Class 1 has 20 students.
Class 2 has 4 fewer students than Class 3.
All four classes together have 68 students.

Unknowns

How many students one grid square (circle) stands for.
The student counts for each class, and the completed graph.

Constraints

Each circle stands for the same number of students.
The four class totals add to 68.

2 · Planchoose the strategy

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

The 4-student difference between Class 2 and Class 3 spans one grid square, which fixes the scale (one circle = 4 students). With the scale known I work out each class in small steps and use the total of 68 to finish.

3 · Execute4 carry out the plan

1Find the value of one grid square

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

Class 3 has 4 more students than Class 2, and that gap is exactly one grid square (one circle). So one circle stands for 4 / 1 = 4 students.

4 \text{ students} \div 1 \text{ circle} = 4 \text{ students per circle}

Matching the known student-difference to the one-circle gap reveals the scale.

2Find Class 1 and Class 3 in circles

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

With 4 students per circle: Class 1 = 20 / 4 = 5 circles, and Class 3 = 20 students = 5 circles. Class 2 is one circle fewer: 4 circles = 16 students.

20 \div 4 = 5,\quad 20 \div 4 = 5

Dividing each known count by the scale converts students to circles.

3Find Class 4 using the total

#7 Identify Subproblems 3.OA.A.3

The four classes total 68. Subtract the three known classes: 68 - 20 - 16 - 20 = 12, so Class 4 has 12 students.

68 - 20 - 16 - 20 = 12

Total minus the known parts gives the missing part.

4Complete the graph

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 4 students per circle: Class 1 = 5, Class 2 = 4, Class 3 = 5, Class 4 = 12 / 4 = 3. Draw circles from the bottom up to those heights.

12 \div 4 = 3

Dividing each count by the scale tells how many circles to draw in each column.

Answer: One square = 4 students; Class 1 = 20 (5 circles), Class 2 = 16 (4 circles), Class 3 = 20 (5 circles), Class 4 = 12 (3 circles).

4 · Reviewdoes it hold up?

Add the completed counts: 20 + 16 + 20 + 12 = 68, matching the total, and Class 2 (16) is exactly 4 fewer than Class 3 (20). Both conditions hold.

Another way: Knowing Class 1 = 20 (5 circles) sets the scale at 20 / 5 = 4 students per circle directly, then the same circle counts follow.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Setting the per-circle scale and drawing each class's circles to complete the scaled picture graph.
3.OA.A.3 Solve multiplication and division word problems within 100 — Using the total with multiplication/subtraction to find Class 4 and convert counts to circles.

💡Takeaway. This only needs the Grade 3 idea that one circle's difference equals the student difference -- that sets the scale!

Set the scale, then complete the graph · 10 practice problems

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