← 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 1/3 of a poster board every 4 minutes. At that same rate, how long does it take her to paint 2/3 of the board?

Givens

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

Unknowns

The number of minutes to paint 2/3 of the board

Constraints

The rate stays the same for every 3th of the board
2/3 is made of 2 copies of the unit fraction 1/3

2 · Planchoose the strategy

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

Seeing 2/3 as 2 units of 1/3 turns the problem into repeated equal chunks: each 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 2/3 as 2 copies of 1/3

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

The fraction 2/3 is 2 copies of the unit fraction 1/3. So painting 2/3 means painting 1/3 2 separate times.

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

A fraction is just a count of unit fractions, so 2/3 is '2 3ths' literally.

2Each 3th costs 4 minutes

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

Painting one 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?

2/3 is 2 times as much board as 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 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 2/3 as 2 copies of the unit fraction 1/3
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/3 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 2/3 takes 2 times the time of one 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 1/5 of a poster board every 3 minutes. At that same rate, how long does it take her to paint 4/5 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 4/5 of the board

Unknowns

The number of minutes to paint 4/5 of the board

Constraints

The rate stays the same for every 5th of the board
4/5 is made of 4 copies of the unit fraction 1/5

2 · Planchoose the strategy

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

Seeing 4/5 as 4 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 4 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See 4/5 as 4 copies of 1/5

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

The fraction 4/5 is 4 copies of the unit fraction 1/5. So painting 4/5 means painting 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 4/5 is '4 5ths' literally.

2Each 5th costs 3 minutes

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

Painting one 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?

4/5 is 4 times as much board as 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 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 4/5 as 4 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

💡Takeaway. A fraction is just a count of unit fractions, so 4/5 takes 4 times the time of one 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 1/4 of a poster board every 5 minutes. At that same rate, how long does it take her to paint 3/4 of the board?

Givens

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

Unknowns

The number of minutes to paint 3/4 of the board

Constraints

The rate stays the same for every 4th of the board
3/4 is made of 3 copies of the unit fraction 1/4

2 · Planchoose the strategy

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

Seeing 3/4 as 3 units of 1/4 turns the problem into repeated equal chunks: each 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 3/4 as 3 copies of 1/4

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

The fraction 3/4 is 3 copies of the unit fraction 1/4. So painting 3/4 means painting 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 3/4 is '3 4ths' literally.

2Each 4th costs 5 minutes

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

Painting one 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?

3/4 is 3 times as much board as 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 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 3/4 as 3 copies of the unit fraction 1/4
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/4 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 3/4 takes 3 times the time of one 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 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 5th of the board
3/5 is made of 3 copies of the unit fraction 1/5

2 · Planchoose the strategy

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

Seeing 3/5 as 3 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 3 times. A bar drawing makes the equal chunks obvious.

3 · Execute2 carry out the plan

1See 3/5 as 3 copies of 1/5

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

The fraction 3/5 is 3 copies of the unit fraction 1/5. So painting 3/5 means painting 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 3/5 is '3 5ths' literally.

2Each 5th costs 3 minutes

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

Painting one 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?

3/5 is 3 times as much board as 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 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 3/5 as 3 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

💡Takeaway. A fraction is just a count of unit fractions, so 3/5 takes 3 times the time of one 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 1/6 of a poster board every 2 minutes. At that same rate, how long does it take her to paint 5/6 of the board?

Givens

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

Unknowns

The number of minutes to paint 5/6 of the board

Constraints

The rate stays the same for every 6th of the board
5/6 is made of 5 copies of the unit fraction 1/6

2 · Planchoose the strategy

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

Seeing 5/6 as 5 units of 1/6 turns the problem into repeated equal chunks: each 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 5/6 as 5 copies of 1/6

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

The fraction 5/6 is 5 copies of the unit fraction 1/6. So painting 5/6 means painting 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 5/6 is '5 6ths' literally.

2Each 6th costs 2 minutes

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

Painting one 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?

5/6 is 5 times as much board as 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 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 5/6 as 5 copies of the unit fraction 1/6
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/6 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 5/6 takes 5 times the time of one 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 1/5 of a poster board every 7 minutes. At that same rate, how long does it take her to paint 2/5 of the board?

Givens

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

Unknowns

The number of minutes to paint 2/5 of the board

Constraints

The rate stays the same for every 5th of the board
2/5 is made of 2 copies of the unit fraction 1/5

2 · Planchoose the strategy

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

Seeing 2/5 as 2 units of 1/5 turns the problem into repeated equal chunks: each 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 2/5 as 2 copies of 1/5

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

The fraction 2/5 is 2 copies of the unit fraction 1/5. So painting 2/5 means painting 1/5 2 separate times.

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

A fraction is just a count of unit fractions, so 2/5 is '2 5ths' literally.

2Each 5th costs 7 minutes

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

Painting one 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?

2/5 is 2 times as much board as 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 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 2/5 as 2 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

💡Takeaway. A fraction is just a count of unit fractions, so 2/5 takes 2 times the time of one 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 1/7 of a poster board every 5 minutes. At that same rate, how long does it take her to paint 6/7 of the board?

Givens

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

Unknowns

The number of minutes to paint 6/7 of the board

Constraints

The rate stays the same for every 7th of the board
6/7 is made of 6 copies of the unit fraction 1/7

2 · Planchoose the strategy

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

Seeing 6/7 as 6 units of 1/7 turns the problem into repeated equal chunks: each 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 6/7 as 6 copies of 1/7

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

The fraction 6/7 is 6 copies of the unit fraction 1/7. So painting 6/7 means painting 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 6/7 is '6 7ths' literally.

2Each 7th costs 5 minutes

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

Painting one 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?

6/7 is 6 times as much board as 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 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 6/7 as 6 copies of the unit fraction 1/7
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/7 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 6/7 takes 6 times the time of one 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 1/7 of a poster board every 6 minutes. At that same rate, how long does it take her to paint 2/7 of the board?

Givens

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

Unknowns

The number of minutes to paint 2/7 of the board

Constraints

The rate stays the same for every 7th of the board
2/7 is made of 2 copies of the unit fraction 1/7

2 · Planchoose the strategy

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

Seeing 2/7 as 2 units of 1/7 turns the problem into repeated equal chunks: each 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 2/7 as 2 copies of 1/7

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

The fraction 2/7 is 2 copies of the unit fraction 1/7. So painting 2/7 means painting 1/7 2 separate times.

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

A fraction is just a count of unit fractions, so 2/7 is '2 7ths' literally.

2Each 7th costs 6 minutes

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

Painting one 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?

2/7 is 2 times as much board as 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 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 2/7 as 2 copies of the unit fraction 1/7
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/7 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 2/7 takes 2 times the time of one 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 1/8 of a poster board every 4 minutes. At that same rate, how long does it take her to paint 5/8 of the board?

Givens

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

Unknowns

The number of minutes to paint 5/8 of the board

Constraints

The rate stays the same for every 8th of the board
5/8 is made of 5 copies of the unit fraction 1/8

2 · Planchoose the strategy

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

Seeing 5/8 as 5 units of 1/8 turns the problem into repeated equal chunks: each 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 5/8 as 5 copies of 1/8

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

The fraction 5/8 is 5 copies of the unit fraction 1/8. So painting 5/8 means painting 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 5/8 is '5 8ths' literally.

2Each 8th costs 4 minutes

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

Painting one 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?

5/8 is 5 times as much board as 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 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 5/8 as 5 copies of the unit fraction 1/8
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/8 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 5/8 takes 5 times the time of one 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 1/4 of a poster board every 8 minutes. At that same rate, how long does it take her to paint 3/4 of the board?

Givens

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

Unknowns

The number of minutes to paint 3/4 of the board

Constraints

The rate stays the same for every 4th of the board
3/4 is made of 3 copies of the unit fraction 1/4

2 · Planchoose the strategy

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

Seeing 3/4 as 3 units of 1/4 turns the problem into repeated equal chunks: each 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 3/4 as 3 copies of 1/4

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

The fraction 3/4 is 3 copies of the unit fraction 1/4. So painting 3/4 means painting 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 3/4 is '3 4ths' literally.

2Each 4th costs 8 minutes

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

Painting one 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?

3/4 is 3 times as much board as 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 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 3/4 as 3 copies of the unit fraction 1/4
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/4 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 3/4 takes 3 times the time of one 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 1/9 of a poster board every 3 minutes. At that same rate, how long does it take her to paint 7/9 of the board?

Givens

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

Unknowns

The number of minutes to paint 7/9 of the board

Constraints

The rate stays the same for every 9th of the board
7/9 is made of 7 copies of the unit fraction 1/9

2 · Planchoose the strategy

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

Seeing 7/9 as 7 units of 1/9 turns the problem into repeated equal chunks: each 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 7/9 as 7 copies of 1/9

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

The fraction 7/9 is 7 copies of the unit fraction 1/9. So painting 7/9 means painting 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 7/9 is '7 9ths' literally.

2Each 9th costs 3 minutes

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

Painting one 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?

7/9 is 7 times as much board as 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 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 7/9 as 7 copies of the unit fraction 1/9
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/9 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 7/9 takes 7 times the time of one 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 1/10 of a poster board every 2 minutes. At that same rate, how long does it take her to paint 3/10 of the board?

Givens

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

Unknowns

The number of minutes to paint 3/10 of the board

Constraints

The rate stays the same for every 10th of the board
3/10 is made of 3 copies of the unit fraction 1/10

2 · Planchoose the strategy

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

Seeing 3/10 as 3 units of 1/10 turns the problem into repeated equal chunks: each 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 3/10 as 3 copies of 1/10

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

The fraction 3/10 is 3 copies of the unit fraction 1/10. So painting 3/10 means painting 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 3/10 is '3 10ths' literally.

2Each 10th costs 2 minutes

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

Painting one 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?

3/10 is 3 times as much board as 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 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 3/10 as 3 copies of the unit fraction 1/10
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Treating each 1/10 as one equal part of the board

💡Takeaway. A fraction is just a count of unit fractions, so 3/10 takes 3 times the time of one 1/10!