A fraction is a count of unit fractions

4.NF.B.33.NF.A.1 · adapt · grade 4

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

It takes Mia $3$ minutes to paint $\dfrac{1}{5}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{3}{5}$ of the poster board?

Show solution

Understand

Mia paints 1/5 of a poster board every 3 minutes. At that same rate, how long does it take her to paint 3/5 of the board?

Givens

Painting 1/5 of the board takes 3 minutes
She paints at a constant (same) rate
We want the time to paint 3/5 of the board

Unknowns

The number of minutes to paint 3/5 of the board

Constraints

The rate stays the same for every fifth of the board
3/5 is made of three copies of the unit fraction 1/5

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

Seeing 3/5 as three units of 1/5 turns the problem into repeated equal chunks: each 1/5 costs 3 minutes, so the total is just that time pattern repeated three times. A bar drawing makes the equal chunks obvious.

Execute

#1 Draw a Diagram 4.NF.B.3

The fraction 3/5 is three copies of the unit fraction 1/5. So painting 3/5 means painting 1/5 three separate times.

\dfrac{3}{5} = \dfrac{1}{5} + \dfrac{1}{5} + \dfrac{1}{5}

A fraction is just a count of unit fractions, so 3/5 is 'three fifths' literally.

#5 Look for a Pattern 3.NF.A.1

Painting one 1/5 takes 3 minutes, and there are three of them, so multiply 3 minutes by 3.

3 \times 3 = 9

Equal-size pieces each take the same time, so three pieces take three times as long.

Answer: 9 minutes

Review

3/5 is three times as much board as 1/5, so it should take three times as long as 3 minutes, which is 9 minutes. That is less than the time for the whole board (5 fifths would be 15 minutes), so 9 minutes is reasonable.

Use units/rate reasoning (tool 8): the rate is 3 minutes per 1/5 of a board; multiplying the rate by the 3 fifths needed gives 3 minutes/fifth times 3 fifths = 9 minutes.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing 3/5 as three copies of the unit fraction 1/5
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/5 as one equal part of the board

💡 A fraction is just a count of unit fractions, so 3/5 takes three times the time of one 1/5!