← Constant increase yields a correspondence equation · Generalize a Growing Pattern into a Rule

Constant increase yields a correspondence equation · 12 practice problems

4.OA.C.55.OA.A.25.OA.B.3

Generated variants — 12

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

Variant 1 easy answer: 14 minutes

A showerhead at Hyuna's house puts out $6\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $9\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $93\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 9 L in it and gains 6 L every minute. We need how many minutes it takes to reach 93 L.

Givens

The showerhead adds 6 L per minute.
The tub already holds 9 L before the shower is turned on.
The target amount is 93 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 9 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 9 L, and each minute adds 6 more.

(0, 9), (1, 15), (2, 21), (3, 27)

Litres = 9 + 6 x minutes.

2Remove the 9 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 9 L came from the showerhead, so subtract it before doing anything with the rate.

93 - 9 = 84

The showerhead has to supply 84 L.

3Split the new water into 6-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 6 L, so count how many of those fit into 84 L.

84 \div 6 = 14

It takes 14 minutes.

Answer: 14 minutes

4 · Reviewdoes it hold up?

Run it forwards: 14 minutes at 6 L is 84 L, and 9 + 84 = 93. It matches.

Another way: Dividing 93 by 6 straight away would give the wrong answer, because it treats the starting 9 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 2 easy answer: 6 minutes

A showerhead at Hyuna's house puts out $15\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $7\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $97\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 7 L in it and gains 15 L every minute. We need how many minutes it takes to reach 97 L.

Givens

The showerhead adds 15 L per minute.
The tub already holds 7 L before the shower is turned on.
The target amount is 97 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 7 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 7 L, and each minute adds 15 more.

(0, 7), (1, 22), (2, 37), (3, 52)

Litres = 7 + 15 x minutes.

2Remove the 7 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 7 L came from the showerhead, so subtract it before doing anything with the rate.

97 - 7 = 90

The showerhead has to supply 90 L.

3Split the new water into 15-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 15 L, so count how many of those fit into 90 L.

90 \div 15 = 6

It takes 6 minutes.

Answer: 6 minutes

4 · Reviewdoes it hold up?

Run it forwards: 6 minutes at 15 L is 90 L, and 7 + 90 = 97. It matches.

Another way: Dividing 97 by 15 straight away would give the wrong answer, because it treats the starting 7 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 3 easy answer: 12 minutes

A showerhead at Hyuna's house puts out $8\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $3\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $99\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 3 L in it and gains 8 L every minute. We need how many minutes it takes to reach 99 L.

Givens

The showerhead adds 8 L per minute.
The tub already holds 3 L before the shower is turned on.
The target amount is 99 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 3 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 3 L, and each minute adds 8 more.

(0, 3), (1, 11), (2, 19), (3, 27)

Litres = 3 + 8 x minutes.

2Remove the 3 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 3 L came from the showerhead, so subtract it before doing anything with the rate.

99 - 3 = 96

The showerhead has to supply 96 L.

3Split the new water into 8-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 8 L, so count how many of those fit into 96 L.

96 \div 8 = 12

It takes 12 minutes.

Answer: 12 minutes

4 · Reviewdoes it hold up?

Run it forwards: 12 minutes at 8 L is 96 L, and 3 + 96 = 99. It matches.

Another way: Dividing 99 by 8 straight away would give the wrong answer, because it treats the starting 3 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 4 easy answer: 9 minutes

A showerhead at Hyuna's house puts out $11\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $13\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $112\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 13 L in it and gains 11 L every minute. We need how many minutes it takes to reach 112 L.

Givens

The showerhead adds 11 L per minute.
The tub already holds 13 L before the shower is turned on.
The target amount is 112 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 13 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 13 L, and each minute adds 11 more.

(0, 13), (1, 24), (2, 35), (3, 46)

Litres = 13 + 11 x minutes.

2Remove the 13 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 13 L came from the showerhead, so subtract it before doing anything with the rate.

112 - 13 = 99

The showerhead has to supply 99 L.

3Split the new water into 11-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 11 L, so count how many of those fit into 99 L.

99 \div 11 = 9

It takes 9 minutes.

Answer: 9 minutes

4 · Reviewdoes it hold up?

Run it forwards: 9 minutes at 11 L is 99 L, and 13 + 99 = 112. It matches.

Another way: Dividing 112 by 11 straight away would give the wrong answer, because it treats the starting 13 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 5 medium answer: 9 minutes

A showerhead at Hyuna's house puts out $12\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $5\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $113\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 5 L in it and gains 12 L every minute. We need how many minutes it takes to reach 113 L.

Givens

The showerhead adds 12 L per minute.
The tub already holds 5 L before the shower is turned on.
The target amount is 113 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 5 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 5 L, and each minute adds 12 more.

(0, 5), (1, 17), (2, 29), (3, 41)

Litres = 5 + 12 x minutes.

2Remove the 5 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 5 L came from the showerhead, so subtract it before doing anything with the rate.

113 - 5 = 108

The showerhead has to supply 108 L.

3Split the new water into 12-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 12 L, so count how many of those fit into 108 L.

108 \div 12 = 9

It takes 9 minutes.

Answer: 9 minutes

4 · Reviewdoes it hold up?

Run it forwards: 9 minutes at 12 L is 108 L, and 5 + 108 = 113. It matches.

Another way: Dividing 113 by 12 straight away would give the wrong answer, because it treats the starting 5 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 6 medium answer: 11 minutes

A showerhead at Hyuna's house puts out $10\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $4\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $114\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 4 L in it and gains 10 L every minute. We need how many minutes it takes to reach 114 L.

Givens

The showerhead adds 10 L per minute.
The tub already holds 4 L before the shower is turned on.
The target amount is 114 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 4 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 4 L, and each minute adds 10 more.

(0, 4), (1, 14), (2, 24), (3, 34)

Litres = 4 + 10 x minutes.

2Remove the 4 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 4 L came from the showerhead, so subtract it before doing anything with the rate.

114 - 4 = 110

The showerhead has to supply 110 L.

3Split the new water into 10-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 10 L, so count how many of those fit into 110 L.

110 \div 10 = 11

It takes 11 minutes.

Answer: 11 minutes

4 · Reviewdoes it hold up?

Run it forwards: 11 minutes at 10 L is 110 L, and 4 + 110 = 114. It matches.

Another way: Dividing 114 by 10 straight away would give the wrong answer, because it treats the starting 4 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 7 medium answer: 15 minutes

A showerhead at Hyuna's house puts out $7\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $11\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $116\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 11 L in it and gains 7 L every minute. We need how many minutes it takes to reach 116 L.

Givens

The showerhead adds 7 L per minute.
The tub already holds 11 L before the shower is turned on.
The target amount is 116 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 11 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 11 L, and each minute adds 7 more.

(0, 11), (1, 18), (2, 25), (3, 32)

Litres = 11 + 7 x minutes.

2Remove the 11 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 11 L came from the showerhead, so subtract it before doing anything with the rate.

116 - 11 = 105

The showerhead has to supply 105 L.

3Split the new water into 7-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 7 L, so count how many of those fit into 105 L.

105 \div 7 = 15

It takes 15 minutes.

Answer: 15 minutes

4 · Reviewdoes it hold up?

Run it forwards: 15 minutes at 7 L is 105 L, and 11 + 105 = 116. It matches.

Another way: Dividing 116 by 7 straight away would give the wrong answer, because it treats the starting 11 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 8 medium answer: 13 minutes

A showerhead at Hyuna's house puts out $9\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $6\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $123\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 6 L in it and gains 9 L every minute. We need how many minutes it takes to reach 123 L.

Givens

The showerhead adds 9 L per minute.
The tub already holds 6 L before the shower is turned on.
The target amount is 123 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 6 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 6 L, and each minute adds 9 more.

(0, 6), (1, 15), (2, 24), (3, 33)

Litres = 6 + 9 x minutes.

2Remove the 6 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 6 L came from the showerhead, so subtract it before doing anything with the rate.

123 - 6 = 117

The showerhead has to supply 117 L.

3Split the new water into 9-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 9 L, so count how many of those fit into 117 L.

117 \div 9 = 13

It takes 13 minutes.

Answer: 13 minutes

4 · Reviewdoes it hold up?

Run it forwards: 13 minutes at 9 L is 117 L, and 6 + 117 = 123. It matches.

Another way: Dividing 123 by 9 straight away would give the wrong answer, because it treats the starting 6 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 9 hard answer: 10 minutes

A showerhead at Hyuna's house puts out $14\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $8\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $148\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 8 L in it and gains 14 L every minute. We need how many minutes it takes to reach 148 L.

Givens

The showerhead adds 14 L per minute.
The tub already holds 8 L before the shower is turned on.
The target amount is 148 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 8 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 8 L, and each minute adds 14 more.

(0, 8), (1, 22), (2, 36), (3, 50)

Litres = 8 + 14 x minutes.

2Remove the 8 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 8 L came from the showerhead, so subtract it before doing anything with the rate.

148 - 8 = 140

The showerhead has to supply 140 L.

3Split the new water into 14-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 14 L, so count how many of those fit into 140 L.

140 \div 14 = 10

It takes 10 minutes.

Answer: 10 minutes

4 · Reviewdoes it hold up?

Run it forwards: 10 minutes at 14 L is 140 L, and 8 + 140 = 148. It matches.

Another way: Dividing 148 by 14 straight away would give the wrong answer, because it treats the starting 8 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 10 hard answer: 8 minutes

A showerhead at Hyuna's house puts out $18\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $5\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $149\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 5 L in it and gains 18 L every minute. We need how many minutes it takes to reach 149 L.

Givens

The showerhead adds 18 L per minute.
The tub already holds 5 L before the shower is turned on.
The target amount is 149 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 5 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 5 L, and each minute adds 18 more.

(0, 5), (1, 23), (2, 41), (3, 59)

Litres = 5 + 18 x minutes.

2Remove the 5 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 5 L came from the showerhead, so subtract it before doing anything with the rate.

149 - 5 = 144

The showerhead has to supply 144 L.

3Split the new water into 18-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 18 L, so count how many of those fit into 144 L.

144 \div 18 = 8

It takes 8 minutes.

Answer: 8 minutes

4 · Reviewdoes it hold up?

Run it forwards: 8 minutes at 18 L is 144 L, and 5 + 144 = 149. It matches.

Another way: Dividing 149 by 18 straight away would give the wrong answer, because it treats the starting 5 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 11 hard answer: 7 minutes

A showerhead at Hyuna's house puts out $20\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $12\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $152\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 12 L in it and gains 20 L every minute. We need how many minutes it takes to reach 152 L.

Givens

The showerhead adds 20 L per minute.
The tub already holds 12 L before the shower is turned on.
The target amount is 152 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 12 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 12 L, and each minute adds 20 more.

(0, 12), (1, 32), (2, 52), (3, 72)

Litres = 12 + 20 x minutes.

2Remove the 12 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 12 L came from the showerhead, so subtract it before doing anything with the rate.

152 - 12 = 140

The showerhead has to supply 140 L.

3Split the new water into 20-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 20 L, so count how many of those fit into 140 L.

140 \div 20 = 7

It takes 7 minutes.

Answer: 7 minutes

4 · Reviewdoes it hold up?

Run it forwards: 7 minutes at 20 L is 140 L, and 12 + 140 = 152. It matches.

Another way: Dividing 152 by 20 straight away would give the wrong answer, because it treats the starting 12 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.

Variant 12 hard answer: 6 minutes

A showerhead at Hyuna's house puts out $25\,\text{L}$ of water each minute. Hyuna wants to use the showerhead to add more water to a bathtub that already holds $10\,\text{L}$ of water. After how many minutes of running the showerhead will the bathtub hold $160\,\text{L}$ of water?

Show solution

1 · Understandwhat's really being asked

A bathtub starts with 10 L in it and gains 25 L every minute. We need how many minutes it takes to reach 160 L.

Givens

The showerhead adds 25 L per minute.
The tub already holds 10 L before the shower is turned on.
The target amount is 160 L.

Unknowns

The number of minutes the showerhead runs.

Constraints

The 10 L was there at the start; the showerhead did not put it there.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #11 Work Backwards#8 Analyze the Units

Build a short minutes-to-litres table to see the rule, then take the head start off the target so what is left is pure showerhead water, which divides cleanly by the rate.

3 · Execute3 carry out the plan

1Build the pattern (minutes to liters)

#5 Look for a Pattern 4.OA.C.5

At 0 minutes the tub holds 10 L, and each minute adds 25 more.

(0, 10), (1, 35), (2, 60), (3, 85)

Litres = 10 + 25 x minutes.

2Remove the 10 L head start

#11 Work Backwards 5.OA.A.2

Only the water above 10 L came from the showerhead, so subtract it before doing anything with the rate.

160 - 10 = 150

The showerhead has to supply 150 L.

3Split the new water into 25-L minutes

#8 Analyze the Units 5.OA.B.3

Each minute is worth 25 L, so count how many of those fit into 150 L.

150 \div 25 = 6

It takes 6 minutes.

Answer: 6 minutes

4 · Reviewdoes it hold up?

Run it forwards: 6 minutes at 25 L is 150 L, and 10 + 150 = 160. It matches.

Another way: Dividing 160 by 25 straight away would give the wrong answer, because it treats the starting 10 L as if the showerhead had poured it. The head start comes off first.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the minutes-to-litres pattern to see the rule.
5.OA.A.2 Write simple expressions that record calculations with numbers — Recording the rule as a starting amount plus a rate times time.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Pairing the minute count with the litre count to find the match.

💡Takeaway. When something starts part-full, take the head start off before you divide -- the rate only explains the part that was added.