← A fraction is a count of unit fractions · Part-Whole Fraction Reasoning

A fraction is a count of unit fractions · 12 practice problems

4.NF.B.33.NF.A.1

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 8 minutes

It takes Mia $4$ minutes to paint $\dfrac{1}{3}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{2}{3}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{3}$ of a poster board every 4 minutes. At that same rate, how long does it take her to paint $\frac{2}{3}$ of the board?

Givens

Painting $\frac{1}{3}$ of the board takes 4 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{2}{3}$ of the board

Unknowns

The number of minutes to paint $\frac{2}{3}$ of the board

Constraints

The rate stays the same for every 3th of the board
$\frac{2}{3}$ is made of 2 copies of the unit fraction $\frac{1}{3}$

2 · Planchoose the strategy

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

Seeing $\frac{2}{3}$ as 2 units of $\frac{1}{3}$ turns the problem into repeated equal chunks: each $\frac{1}{3}$ costs 4 minutes, so the total is just that time pattern repeated 2 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{2}{3}$ as 2 copies of $\frac{1}{3}$

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

The fraction

\frac{2}{3}

is 2 copies of the unit fraction

\frac{1}{3}

. So painting

\frac{2}{3}

means painting

\frac{1}{3}

2 separate times.

\dfrac{2}{3} = \dfrac{1}{3} + \dfrac{1}{3}

A fraction is just a count of unit fractions, so

\frac{2}{3}

is '2 3ths' literally.

2Each 3th costs 4 minutes

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

Painting one

\frac{1}{3}

takes 4 minutes, and there are 2 of them, so multiply 4 minutes by 2.

4 \times 2 = 8

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 8 minutes

4 · Reviewdoes it hold up?

$\frac{2}{3}$ is 2 times as much board as $\frac{1}{3}$ , so it should take 2 times as long as 4 minutes, which is 8 minutes. That is less than the time for the whole board (3 3ths would be 12 minutes), so 8 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 4 minutes per $\frac{1}{3}$ of a board; multiplying the rate by the 2 3ths needed gives 4 minutes/3th times 2 3ths = 8 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{2}{3}$ as 2 copies of the unit fraction $\frac{1}{3}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{3}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{2}{3}

takes 2 times the time of one

\frac{1}{3}

Variant 2 easy answer: 12 minutes

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 $\dfrac{4}{5}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{5}$ of a poster board every 3 minutes. At that same rate, how long does it take her to paint $\frac{4}{5}$ of the board?

Givens

Painting $\frac{1}{5}$ of the board takes 3 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{4}{5}$ of the board

Unknowns

The number of minutes to paint $\frac{4}{5}$ of the board

Constraints

The rate stays the same for every 5th of the board
$\frac{4}{5}$ is made of 4 copies of the unit fraction $\frac{1}{5}$

2 · Planchoose the strategy

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

Seeing $\frac{4}{5}$ as 4 units of $\frac{1}{5}$ turns the problem into repeated equal chunks: each $\frac{1}{5}$ costs 3 minutes, so the total is just that time pattern repeated 4 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{4}{5}$ as 4 copies of $\frac{1}{5}$

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

The fraction

\frac{4}{5}

is 4 copies of the unit fraction

\frac{1}{5}

. So painting

\frac{4}{5}

means painting

\frac{1}{5}

4 separate times.

\dfrac{4}{5} = \dfrac{1}{5} + \dfrac{1}{5} + \dfrac{1}{5} + \dfrac{1}{5}

A fraction is just a count of unit fractions, so

\frac{4}{5}

is '4 5ths' literally.

2Each 5th costs 3 minutes

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

Painting one

\frac{1}{5}

takes 3 minutes, and there are 4 of them, so multiply 3 minutes by 4.

3 \times 4 = 12

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 12 minutes

4 · Reviewdoes it hold up?

$\frac{4}{5}$ is 4 times as much board as $\frac{1}{5}$ , so it should take 4 times as long as 3 minutes, which is 12 minutes. That is less than the time for the whole board (5 5ths would be 15 minutes), so 12 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 3 minutes per $\frac{1}{5}$ of a board; multiplying the rate by the 4 5ths needed gives 3 minutes/5th times 4 5ths = 12 minutes.

Standardsmin grade 4

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

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{4}{5}

takes 4 times the time of one

\frac{1}{5}

Variant 3 easy answer: 15 minutes

It takes Mia $5$ minutes to paint $\dfrac{1}{4}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{3}{4}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{4}$ of a poster board every 5 minutes. At that same rate, how long does it take her to paint $\frac{3}{4}$ of the board?

Givens

Painting $\frac{1}{4}$ of the board takes 5 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{3}{4}$ of the board

Unknowns

The number of minutes to paint $\frac{3}{4}$ of the board

Constraints

The rate stays the same for every 4th of the board
$\frac{3}{4}$ is made of 3 copies of the unit fraction $\frac{1}{4}$

2 · Planchoose the strategy

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

Seeing $\frac{3}{4}$ as 3 units of $\frac{1}{4}$ turns the problem into repeated equal chunks: each $\frac{1}{4}$ costs 5 minutes, so the total is just that time pattern repeated 3 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{3}{4}$ as 3 copies of $\frac{1}{4}$

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

The fraction

\frac{3}{4}

is 3 copies of the unit fraction

\frac{1}{4}

. So painting

\frac{3}{4}

means painting

\frac{1}{4}

3 separate times.

\dfrac{3}{4} = \dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4}

A fraction is just a count of unit fractions, so

\frac{3}{4}

is '3 4ths' literally.

2Each 4th costs 5 minutes

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

Painting one

\frac{1}{4}

takes 5 minutes, and there are 3 of them, so multiply 5 minutes by 3.

5 \times 3 = 15

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 15 minutes

4 · Reviewdoes it hold up?

$\frac{3}{4}$ is 3 times as much board as $\frac{1}{4}$ , so it should take 3 times as long as 5 minutes, which is 15 minutes. That is less than the time for the whole board (4 4ths would be 20 minutes), so 15 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 5 minutes per $\frac{1}{4}$ of a board; multiplying the rate by the 3 4ths needed gives 5 minutes/4th times 3 4ths = 15 minutes.

Standardsmin grade 4

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

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{3}{4}

takes 3 times the time of one

\frac{1}{4}

Variant 4 easy answer: 9 minutes

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

1 · Understandwhat's really being asked

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

Givens

Painting $\frac{1}{5}$ of the board takes 3 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{3}{5}$ of the board

Unknowns

The number of minutes to paint $\frac{3}{5}$ of the board

Constraints

The rate stays the same for every 5th of the board
$\frac{3}{5}$ is made of 3 copies of the unit fraction $\frac{1}{5}$

2 · Planchoose the strategy

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

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

3 · Execute2 carry out the plan

1See $\frac{3}{5}$ as 3 copies of $\frac{1}{5}$

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

The fraction

\frac{3}{5}

is 3 copies of the unit fraction

\frac{1}{5}

. So painting

\frac{3}{5}

means painting

\frac{1}{5}

3 separate times.

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

A fraction is just a count of unit fractions, so

\frac{3}{5}

is '3 5ths' literally.

2Each 5th costs 3 minutes

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

Painting one

\frac{1}{5}

takes 3 minutes, and there are 3 of them, so multiply 3 minutes by 3.

3 \times 3 = 9

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 9 minutes

4 · Reviewdoes it hold up?

$\frac{3}{5}$ is 3 times as much board as $\frac{1}{5}$ , so it should take 3 times as long as 3 minutes, which is 9 minutes. That is less than the time for the whole board (5 5ths would be 15 minutes), so 9 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 3 minutes per $\frac{1}{5}$ of a board; multiplying the rate by the 3 5ths needed gives 3 minutes/5th times 3 5ths = 9 minutes.

Standardsmin grade 4

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

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{3}{5}

takes 3 times the time of one

\frac{1}{5}

Variant 5 medium answer: 10 minutes

It takes Mia $2$ minutes to paint $\dfrac{1}{6}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{5}{6}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{6}$ of a poster board every 2 minutes. At that same rate, how long does it take her to paint $\frac{5}{6}$ of the board?

Givens

Painting $\frac{1}{6}$ of the board takes 2 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{5}{6}$ of the board

Unknowns

The number of minutes to paint $\frac{5}{6}$ of the board

Constraints

The rate stays the same for every 6th of the board
$\frac{5}{6}$ is made of 5 copies of the unit fraction $\frac{1}{6}$

2 · Planchoose the strategy

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

Seeing $\frac{5}{6}$ as 5 units of $\frac{1}{6}$ turns the problem into repeated equal chunks: each $\frac{1}{6}$ costs 2 minutes, so the total is just that time pattern repeated 5 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{5}{6}$ as 5 copies of $\frac{1}{6}$

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

The fraction

\frac{5}{6}

is 5 copies of the unit fraction

\frac{1}{6}

. So painting

\frac{5}{6}

means painting

\frac{1}{6}

5 separate times.

\dfrac{5}{6} = \dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6}

A fraction is just a count of unit fractions, so

\frac{5}{6}

is '5 6ths' literally.

2Each 6th costs 2 minutes

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

Painting one

\frac{1}{6}

takes 2 minutes, and there are 5 of them, so multiply 2 minutes by 5.

2 \times 5 = 10

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 10 minutes

4 · Reviewdoes it hold up?

$\frac{5}{6}$ is 5 times as much board as $\frac{1}{6}$ , so it should take 5 times as long as 2 minutes, which is 10 minutes. That is less than the time for the whole board (6 6ths would be 12 minutes), so 10 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 2 minutes per $\frac{1}{6}$ of a board; multiplying the rate by the 5 6ths needed gives 2 minutes/6th times 5 6ths = 10 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{5}{6}$ as 5 copies of the unit fraction $\frac{1}{6}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{6}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{5}{6}

takes 5 times the time of one

\frac{1}{6}

Variant 6 medium answer: 14 minutes

It takes Mia $7$ minutes to paint $\dfrac{1}{5}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{2}{5}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{5}$ of a poster board every 7 minutes. At that same rate, how long does it take her to paint $\frac{2}{5}$ of the board?

Givens

Painting $\frac{1}{5}$ of the board takes 7 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{2}{5}$ of the board

Unknowns

The number of minutes to paint $\frac{2}{5}$ of the board

Constraints

The rate stays the same for every 5th of the board
$\frac{2}{5}$ is made of 2 copies of the unit fraction $\frac{1}{5}$

2 · Planchoose the strategy

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

Seeing $\frac{2}{5}$ as 2 units of $\frac{1}{5}$ turns the problem into repeated equal chunks: each $\frac{1}{5}$ costs 7 minutes, so the total is just that time pattern repeated 2 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{2}{5}$ as 2 copies of $\frac{1}{5}$

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

The fraction

\frac{2}{5}

is 2 copies of the unit fraction

\frac{1}{5}

. So painting

\frac{2}{5}

means painting

\frac{1}{5}

2 separate times.

\dfrac{2}{5} = \dfrac{1}{5} + \dfrac{1}{5}

A fraction is just a count of unit fractions, so

\frac{2}{5}

is '2 5ths' literally.

2Each 5th costs 7 minutes

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

Painting one

\frac{1}{5}

takes 7 minutes, and there are 2 of them, so multiply 7 minutes by 2.

7 \times 2 = 14

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 14 minutes

4 · Reviewdoes it hold up?

$\frac{2}{5}$ is 2 times as much board as $\frac{1}{5}$ , so it should take 2 times as long as 7 minutes, which is 14 minutes. That is less than the time for the whole board (5 5ths would be 35 minutes), so 14 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 7 minutes per $\frac{1}{5}$ of a board; multiplying the rate by the 2 5ths needed gives 7 minutes/5th times 2 5ths = 14 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{2}{5}$ as 2 copies of the unit fraction $\frac{1}{5}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{5}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{2}{5}

takes 2 times the time of one

\frac{1}{5}

Variant 7 medium answer: 30 minutes

It takes Mia $5$ minutes to paint $\dfrac{1}{7}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{6}{7}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{7}$ of a poster board every 5 minutes. At that same rate, how long does it take her to paint $\frac{6}{7}$ of the board?

Givens

Painting $\frac{1}{7}$ of the board takes 5 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{6}{7}$ of the board

Unknowns

The number of minutes to paint $\frac{6}{7}$ of the board

Constraints

The rate stays the same for every 7th of the board
$\frac{6}{7}$ is made of 6 copies of the unit fraction $\frac{1}{7}$

2 · Planchoose the strategy

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

Seeing $\frac{6}{7}$ as 6 units of $\frac{1}{7}$ turns the problem into repeated equal chunks: each $\frac{1}{7}$ costs 5 minutes, so the total is just that time pattern repeated 6 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{6}{7}$ as 6 copies of $\frac{1}{7}$

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

The fraction

\frac{6}{7}

is 6 copies of the unit fraction

\frac{1}{7}

. So painting

\frac{6}{7}

means painting

\frac{1}{7}

6 separate times.

\dfrac{6}{7} = \dfrac{1}{7} + \dfrac{1}{7} + \dfrac{1}{7} + \dfrac{1}{7} + \dfrac{1}{7} + \dfrac{1}{7}

A fraction is just a count of unit fractions, so

\frac{6}{7}

is '6 7ths' literally.

2Each 7th costs 5 minutes

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

Painting one

\frac{1}{7}

takes 5 minutes, and there are 6 of them, so multiply 5 minutes by 6.

5 \times 6 = 30

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 30 minutes

4 · Reviewdoes it hold up?

$\frac{6}{7}$ is 6 times as much board as $\frac{1}{7}$ , so it should take 6 times as long as 5 minutes, which is 30 minutes. That is less than the time for the whole board (7 7ths would be 35 minutes), so 30 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 5 minutes per $\frac{1}{7}$ of a board; multiplying the rate by the 6 7ths needed gives 5 minutes/7th times 6 7ths = 30 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{6}{7}$ as 6 copies of the unit fraction $\frac{1}{7}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{7}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{6}{7}

takes 6 times the time of one

\frac{1}{7}

Variant 8 medium answer: 12 minutes

It takes Mia $6$ minutes to paint $\dfrac{1}{7}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{2}{7}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{7}$ of a poster board every 6 minutes. At that same rate, how long does it take her to paint $\frac{2}{7}$ of the board?

Givens

Painting $\frac{1}{7}$ of the board takes 6 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{2}{7}$ of the board

Unknowns

The number of minutes to paint $\frac{2}{7}$ of the board

Constraints

The rate stays the same for every 7th of the board
$\frac{2}{7}$ is made of 2 copies of the unit fraction $\frac{1}{7}$

2 · Planchoose the strategy

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

Seeing $\frac{2}{7}$ as 2 units of $\frac{1}{7}$ turns the problem into repeated equal chunks: each $\frac{1}{7}$ costs 6 minutes, so the total is just that time pattern repeated 2 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{2}{7}$ as 2 copies of $\frac{1}{7}$

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

The fraction

\frac{2}{7}

is 2 copies of the unit fraction

\frac{1}{7}

. So painting

\frac{2}{7}

means painting

\frac{1}{7}

2 separate times.

\dfrac{2}{7} = \dfrac{1}{7} + \dfrac{1}{7}

A fraction is just a count of unit fractions, so

\frac{2}{7}

is '2 7ths' literally.

2Each 7th costs 6 minutes

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

Painting one

\frac{1}{7}

takes 6 minutes, and there are 2 of them, so multiply 6 minutes by 2.

6 \times 2 = 12

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 12 minutes

4 · Reviewdoes it hold up?

$\frac{2}{7}$ is 2 times as much board as $\frac{1}{7}$ , so it should take 2 times as long as 6 minutes, which is 12 minutes. That is less than the time for the whole board (7 7ths would be 42 minutes), so 12 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 6 minutes per $\frac{1}{7}$ of a board; multiplying the rate by the 2 7ths needed gives 6 minutes/7th times 2 7ths = 12 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{2}{7}$ as 2 copies of the unit fraction $\frac{1}{7}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{7}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{2}{7}

takes 2 times the time of one

\frac{1}{7}

Variant 9 hard answer: 20 minutes

It takes Mia $4$ minutes to paint $\dfrac{1}{8}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{5}{8}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{8}$ of a poster board every 4 minutes. At that same rate, how long does it take her to paint $\frac{5}{8}$ of the board?

Givens

Painting $\frac{1}{8}$ of the board takes 4 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{5}{8}$ of the board

Unknowns

The number of minutes to paint $\frac{5}{8}$ of the board

Constraints

The rate stays the same for every 8th of the board
$\frac{5}{8}$ is made of 5 copies of the unit fraction $\frac{1}{8}$

2 · Planchoose the strategy

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

Seeing $\frac{5}{8}$ as 5 units of $\frac{1}{8}$ turns the problem into repeated equal chunks: each $\frac{1}{8}$ costs 4 minutes, so the total is just that time pattern repeated 5 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{5}{8}$ as 5 copies of $\frac{1}{8}$

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

The fraction

\frac{5}{8}

is 5 copies of the unit fraction

\frac{1}{8}

. So painting

\frac{5}{8}

means painting

\frac{1}{8}

5 separate times.

\dfrac{5}{8} = \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8}

A fraction is just a count of unit fractions, so

\frac{5}{8}

is '5 8ths' literally.

2Each 8th costs 4 minutes

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

Painting one

\frac{1}{8}

takes 4 minutes, and there are 5 of them, so multiply 4 minutes by 5.

4 \times 5 = 20

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 20 minutes

4 · Reviewdoes it hold up?

$\frac{5}{8}$ is 5 times as much board as $\frac{1}{8}$ , so it should take 5 times as long as 4 minutes, which is 20 minutes. That is less than the time for the whole board (8 8ths would be 32 minutes), so 20 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 4 minutes per $\frac{1}{8}$ of a board; multiplying the rate by the 5 8ths needed gives 4 minutes/8th times 5 8ths = 20 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{5}{8}$ as 5 copies of the unit fraction $\frac{1}{8}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{8}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{5}{8}

takes 5 times the time of one

\frac{1}{8}

Variant 10 hard answer: 24 minutes

It takes Mia $8$ minutes to paint $\dfrac{1}{4}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{3}{4}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{4}$ of a poster board every 8 minutes. At that same rate, how long does it take her to paint $\frac{3}{4}$ of the board?

Givens

Painting $\frac{1}{4}$ of the board takes 8 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{3}{4}$ of the board

Unknowns

The number of minutes to paint $\frac{3}{4}$ of the board

Constraints

The rate stays the same for every 4th of the board
$\frac{3}{4}$ is made of 3 copies of the unit fraction $\frac{1}{4}$

2 · Planchoose the strategy

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

Seeing $\frac{3}{4}$ as 3 units of $\frac{1}{4}$ turns the problem into repeated equal chunks: each $\frac{1}{4}$ costs 8 minutes, so the total is just that time pattern repeated 3 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{3}{4}$ as 3 copies of $\frac{1}{4}$

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

The fraction

\frac{3}{4}

is 3 copies of the unit fraction

\frac{1}{4}

. So painting

\frac{3}{4}

means painting

\frac{1}{4}

3 separate times.

\dfrac{3}{4} = \dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4}

A fraction is just a count of unit fractions, so

\frac{3}{4}

is '3 4ths' literally.

2Each 4th costs 8 minutes

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

Painting one

\frac{1}{4}

takes 8 minutes, and there are 3 of them, so multiply 8 minutes by 3.

8 \times 3 = 24

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 24 minutes

4 · Reviewdoes it hold up?

$\frac{3}{4}$ is 3 times as much board as $\frac{1}{4}$ , so it should take 3 times as long as 8 minutes, which is 24 minutes. That is less than the time for the whole board (4 4ths would be 32 minutes), so 24 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 8 minutes per $\frac{1}{4}$ of a board; multiplying the rate by the 3 4ths needed gives 8 minutes/4th times 3 4ths = 24 minutes.

Standardsmin grade 4

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

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{3}{4}

takes 3 times the time of one

\frac{1}{4}

Variant 11 hard answer: 21 minutes

It takes Mia $3$ minutes to paint $\dfrac{1}{9}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{7}{9}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{9}$ of a poster board every 3 minutes. At that same rate, how long does it take her to paint $\frac{7}{9}$ of the board?

Givens

Painting $\frac{1}{9}$ of the board takes 3 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{7}{9}$ of the board

Unknowns

The number of minutes to paint $\frac{7}{9}$ of the board

Constraints

The rate stays the same for every 9th of the board
$\frac{7}{9}$ is made of 7 copies of the unit fraction $\frac{1}{9}$

2 · Planchoose the strategy

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

Seeing $\frac{7}{9}$ as 7 units of $\frac{1}{9}$ turns the problem into repeated equal chunks: each $\frac{1}{9}$ costs 3 minutes, so the total is just that time pattern repeated 7 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{7}{9}$ as 7 copies of $\frac{1}{9}$

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

The fraction

\frac{7}{9}

is 7 copies of the unit fraction

\frac{1}{9}

. So painting

\frac{7}{9}

means painting

\frac{1}{9}

7 separate times.

\dfrac{7}{9} = \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{9}

A fraction is just a count of unit fractions, so

\frac{7}{9}

is '7 9ths' literally.

2Each 9th costs 3 minutes

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

Painting one

\frac{1}{9}

takes 3 minutes, and there are 7 of them, so multiply 3 minutes by 7.

3 \times 7 = 21

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 21 minutes

4 · Reviewdoes it hold up?

$\frac{7}{9}$ is 7 times as much board as $\frac{1}{9}$ , so it should take 7 times as long as 3 minutes, which is 21 minutes. That is less than the time for the whole board (9 9ths would be 27 minutes), so 21 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 3 minutes per $\frac{1}{9}$ of a board; multiplying the rate by the 7 9ths needed gives 3 minutes/9th times 7 9ths = 21 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{7}{9}$ as 7 copies of the unit fraction $\frac{1}{9}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{9}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{7}{9}

takes 7 times the time of one

\frac{1}{9}

Variant 12 hard answer: 6 minutes

It takes Mia $2$ minutes to paint $\dfrac{1}{10}$ of a poster board. Working at the same rate, how many minutes will it take her to paint $\dfrac{3}{10}$ of the poster board?

Show solution

1 · Understandwhat's really being asked

Mia paints $\frac{1}{10}$ of a poster board every 2 minutes. At that same rate, how long does it take her to paint $\frac{3}{10}$ of the board?

Givens

Painting $\frac{1}{10}$ of the board takes 2 minutes
She paints at a constant (same) rate
We want the time to paint $\frac{3}{10}$ of the board

Unknowns

The number of minutes to paint $\frac{3}{10}$ of the board

Constraints

The rate stays the same for every 10th of the board
$\frac{3}{10}$ is made of 3 copies of the unit fraction $\frac{1}{10}$

2 · Planchoose the strategy

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

Seeing $\frac{3}{10}$ as 3 units of $\frac{1}{10}$ turns the problem into repeated equal chunks: each $\frac{1}{10}$ costs 2 minutes, so the total is just that time pattern repeated 3 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See $\frac{3}{10}$ as 3 copies of $\frac{1}{10}$

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

The fraction

\frac{3}{10}

is 3 copies of the unit fraction

\frac{1}{10}

. So painting

\frac{3}{10}

means painting

\frac{1}{10}

3 separate times.

\dfrac{3}{10} = \dfrac{1}{10} + \dfrac{1}{10} + \dfrac{1}{10}

A fraction is just a count of unit fractions, so

\frac{3}{10}

is '3 10ths' literally.

2Each 10th costs 2 minutes

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

Painting one

\frac{1}{10}

takes 2 minutes, and there are 3 of them, so multiply 2 minutes by 3.

2 \times 3 = 6

Equal-size pieces each take the same time, so more pieces take proportionally longer.

Answer: 6 minutes

4 · Reviewdoes it hold up?

$\frac{3}{10}$ is 3 times as much board as $\frac{1}{10}$ , so it should take 3 times as long as 2 minutes, which is 6 minutes. That is less than the time for the whole board (10 10ths would be 20 minutes), so 6 minutes is reasonable.

Another way: Use units/rate reasoning (tool 8): the rate is 2 minutes per $\frac{1}{10}$ of a board; multiplying the rate by the 3 10ths needed gives 2 minutes/10th times 3 10ths = 6 minutes.

Standardsmin grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Seeing $\frac{3}{10}$ as 3 copies of the unit fraction $\frac{1}{10}$
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each $\frac{1}{10}$ as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so

\frac{3}{10}

takes 3 times the time of one

\frac{1}{10}