Set the scale, then complete the graph

3.MD.B.3 · take · grade 3

Archetype: Read and Scale a Data Graph · step in a 21-type progression

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 (○). 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 (○) for each class from the bottom up to complete the graph.

Show solution

Understand

I must complete a 'Students per Class' picture graph for four classes. The grid already shows Class 2 with 3 circles and Class 3 with 4 circles. 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 the missing classes.

Givens

Class 1 has 16 students.
Class 2 has 4 fewer students than Class 3.
All four classes together have 56 students.
On the partly drawn graph, Class 2 has 3 circles and Class 3 has 4 circles (Class 1 and Class 4 are empty).

Unknowns

How many students one grid square (circle) stands for.
The student counts for Class 2, Class 3, and Class 4, and the completed graph.

Constraints

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

Plan

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

The difference of 1 circle between Class 2 and Class 3 must equal the 4-student difference, 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.

Execute

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

Class 3 (4 circles) has 4 more students than Class 2 (3 circles). That difference of 1 circle equals 4 students, so one circle stands for 4 students.

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

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

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

With 4 students per circle: Class 2 = 3 × 4 = 12 students and Class 3 = 4 × 4 = 16 students. (Check: 16 - 12 = 4, the required difference.)

3 \times 4 = 12,\quad 4 \times 4 = 16

Multiplying circles by the scale converts the drawing to real counts.

#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.

#7 Identify Subproblems 3.MD.B.3

Convert each class to circles at 4 students per circle: Class 1 = 16 ÷ 4 = 4 circles, Class 2 = 3 circles, Class 3 = 4 circles, Class 4 = 12 ÷ 4 = 3 circles. Draw circles from the bottom up to those heights.

16 \div 4 = 4,\quad 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).

Review

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.

First find Class 2 + Class 3 = 56 - 16 - 12... instead, knowing Class 1 = 16 (4 circles) sets the scale at 16 ÷ 4 = 4 students per circle directly, then the same circle counts follow.

Standards · min 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 of 56 with multiplication/subtraction to find Class 4 and convert counts to circles.

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