4.2: Fractions from Division
Fractions from Division
Division doesn’t always “work out evenly.” That’s where fractions and decimals appear naturally.
Partition Division with Fractions
Example using Partition Division
Problem: 7 cookies shared among 3 children.
- First, give each child 2 cookies (6 total).
- 1 cookie remains.
- Split that cookie into 3 equal parts.
Each child gets 2 and 1/3 cookies.
So, 7 ÷ 3 = 2 ⅓.
Visual (Partition):
🍪🍪 | 🍪🍪 | 🍪🍪 | leftover 🍪 split → thirds
This interpretation shows fractions as fair shares.
Measurement Division with Fractions
Example using Measurement Division
Problem: 7 cookies placed into groups of 3.
- 2 full groups of 3 can be made.
- 1 cookie is left over.
- That 1 cookie is 1/3 of another group of 3.
So, 7 ÷ 3 = 2 ⅓.
Visual (Measurement):
[🍪🍪🍪] [🍪🍪🍪] [🍪(⅓ group)]
This interpretation shows fractions as a part of a group.
Extending to Decimals
Decimals arise when we continue partitioning into smaller and smaller equal parts.
Examples with Decimals
Example: 7 ÷ 4
- Partition: 7 cookies shared among 4 children.
- Each gets 1 cookie (4 total).
- 3 cookies left → split into fourths.
- Each child gets 1 + ¾ = 1.75 cookies.
- Measurement: 7 cookies into groups of 4.
- 1 full group of 4 fits.
- 3 cookies left.
- 3 ÷ 4 = 0.75 more groups.
- Total groups = 1.75.
Both interpretations agree: 7 ÷ 4 = 1.75.
Classroom Connections
- Use manipulatives (counters, blocks, fraction bars) for both interpretations.
- Ask students to identify whether a word problem is fair share or how many groups.
- Show fractions and decimals as natural outcomes when division doesn’t come out evenly.
- Relate to long division: Measurement division connects to repeated subtraction, while partition division helps interpret the decimal result as fair shares.
Key Takeaway:
Division has two interpretations—partition (fair share) and measurement (repeated subtraction). Both extend naturally into fractions and decimals, giving students a richer understanding of what division means beyond whole numbers.
