Fractions require equal partitioning of the whole

3.NF.A.13.G.A.2 · take · grade 3

Archetype: Part-Whole Fraction Reasoning · step in a 5-type progression

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 orange.

Show solution

Understand

A rectangle is partitioned by tick marks into 32 equal cells (an 8-by-4 grid). Part of it is shaded orange. 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 orange 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

Plan

#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.

Execute

#1 Draw a Diagram 3.G.A.2

The tick marks split the rectangle into an 8-by-4 array of equal cells, so the whole is 32 equal parts and each cell is 1/32.

8 \times 4 = 32

Equal partitioning is what lets us name the shaded amount with a single fraction.

#7 Identify Subproblems 3.NF.A.1

The orange 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 \times 3 = 21 \text{ shaded cells out of } 32

Counting full rows of equal cells is faster than counting one cell at a time.

#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.

\dfrac{21}{32}

The shaded part as a fraction of the whole is just shaded cells over all cells.

Answer: 21/32

Review

The unshaded region is the rightmost column plus the bottom row, which is more than half a column of the picture but well under half the rectangle, so the shaded part should be a bit more than 1/2 of the whole; 21/32 is about 0.66, which matches the picture being mostly orange.

Change focus to the complement (tool 16): the unshaded cells are the 8 bottom-row cells plus the 3 remaining cells of the right column, which is 8 + 3 = 11 cells, so 11/32 is unshaded and the shaded part is 1 - 11/32 = 21/32.

Standards · min 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

💡 When the whole is split into equal cells, the shaded fraction is just shaded cells over all cells!