Find the value of one grid square

3.MD.B.3 · take · grade 3

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

Representative Problem

The bar graph shows the historical figure most admired by the $46$ fourth-grade students at Liam's school. Find how many students admire Lincoln.

Bar graph "Most-Admired Historical Figure": the horizontal axis lists the figures (Lincoln, Washington, Franklin, Tubman) and the vertical axis shows the number of students. The vertical scale has no numbers written on it. The height of each bar, counted in grid squares, is: Lincoln $8$ squares, Washington $7$ squares, Franklin $3$ squares, Tubman $5$ squares.

Adding the number of grid squares for all the bars gives $8 +$ [ ] $+$ [ ] $+$ [ ] $=$ [ ] (squares).
Since [ ] squares represent $46$ students, one grid square represents [ ] students.
Therefore the number of students who admire Lincoln is [ ].

Show solution

Understand

A bar graph shows how many of 46 fourth-graders most admire each of four figures, but the vertical scale has no numbers. The bars are 8, 7, 3, and 5 grid squares tall (Lincoln, Washington, Franklin, Tubman). Find how many students one grid square stands for, then find how many admire Lincoln.

Givens

Total students = 46
Bar heights in grid squares: Lincoln 8, Washington 7, Franklin 3, Tubman 5
The vertical scale has no numbers written on it

Unknowns

How many students one grid square represents
The number of students who admire Lincoln

Constraints

Every grid square represents the same number of students
The four bar values together account for all 46 students

Plan

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

The key is the unit 'students per square'. Total squares correspond to total students, so dividing 46 by the total squares gives the value of one square, which then scales the Lincoln bar.

Execute

#7 Identify Subproblems 3.MD.B.3

Sum the heights of all four bars to get the total number of grid squares used.

8 + 7 + 3 + 5 = 23

All the squares together stand for all the students.

#8 Analyze the Units 3.OA.A.3

Those 23 squares represent all 46 students, so one square represents 46 divided by 23.

46 \div 23 = 2

Sharing 46 students evenly across 23 squares gives 2 per square.

#8 Analyze the Units 3.OA.A.3

Lincoln's bar is 8 squares tall, and each square is worth 2 students, so multiply.

8 \times 2 = 16

8 squares each worth 2 students make 16 students.

Answer: 16 students

Review

Checking all bars at 2 per square: 16 + 14 + 6 + 10 = 46, matching the total, so the value-per-square and the Lincoln count are correct.

Look for a pattern (tool 5): once one square = 2, just double every bar height (8,7,3,5 to 16,14,6,10) and confirm they sum to 46.

Standards · min grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading bar heights in grid squares and relating them to the total
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 46 by 23 for the unit value and multiplying 8 by 2 for Lincoln

💡 This only needs Grade 3 division: share the total over all the squares to learn what one square is worth!