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Largest vertical gap between two line graphs · 8 practice problems

5.MD.B.2

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Generated variants — 8

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

Variant 1 easy

The line graph shows Sofia's and Max's reading counts, recorded over time. Find when the difference between their two values is smallest, and find that difference in books.

(Figure) A line graph titled "Reading Counts" with both data sets drawn on the same axes. The horizontal axis is month ( $1$ , $2$ , $3$ , $4$ , $5$ ). The vertical axis is in books, with major gridlines at $0$ , $5$ , $10$ and each small grid square worth $1$ books (the lower part of the axis is cut off with a wavy line). Sofia's values are $4$ , $7$ , $9$ , $12$ , $14$ books; Max's values are $1$ , $5$ , $8$ , $9$ , $10$ books.

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1 · Understandwhat's really being asked

One line graph shows Sofia's and Max's reading counts at month 1, 2, 3, 4, 5. Sofia: 4, 7, 9, 12, 14 books; Max: 1, 5, 8, 9, 10 books (each small square = 1 books). I must find the month where the two values are closest and give that smallest difference.

Givens

Sofia's values: 1=4, 2=7, 3=9, 4=12, 5=14 books
Max's values: 1=1, 2=5, 3=8, 4=9, 5=10 books
Each small grid square = 1 books
Both lines are drawn on the same axes

Unknowns

The month at which the two values differ the least
That smallest difference in books

Constraints

Compare the two lines only at the marked month values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Sofia minus Max) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Max's value from Sofia's at each category: month 1: 4-1=3; month 2: 7-5=2; month 3: 9-8=1; month 4: 12-9=3; month 5: 14-10=4.

3,\ 2,\ 1,\ 3,\ 4

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 3, 2, 1, 3, 4 the smallest is 1, which happens at month 3. On the graph this is where the two lines are closest.

\min(3,2,1,3,4) = 1\ \text{at month } 3

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At month 3, difference = 1 books

4 · Reviewdoes it hold up?

At month 3 the gap is 1 books, and every other category gives a larger gap, so 3 is correctly the closest. Each square is 1 books.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 2 easy

The line graph shows Jin's and Rae's steps walked, recorded over time. Find when the difference between their two values is smallest, and find that difference in thousand steps.

(Figure) A line graph titled "Steps Walked" with both data sets drawn on the same axes. The horizontal axis is day ( $1$ , $2$ , $3$ , $4$ , $5$ ). The vertical axis is in thousand steps, with major gridlines at $0$ , $8$ , $16$ and each small grid square worth $2$ thousand steps (the lower part of the axis is cut off with a wavy line). Jin's values are $6$ , $9$ , $11$ , $14$ , $16$ thousand steps; Rae's values are $3$ , $7$ , $10$ , $11$ , $14$ thousand steps.

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1 · Understandwhat's really being asked

One line graph shows Jin's and Rae's steps walked at day 1, 2, 3, 4, 5. Jin: 6, 9, 11, 14, 16 thousand steps; Rae: 3, 7, 10, 11, 14 thousand steps (each small square = 2 thousand steps). I must find the day where the two values are closest and give that smallest difference.

Givens

Jin's values: 1=6, 2=9, 3=11, 4=14, 5=16 thousand steps
Rae's values: 1=3, 2=7, 3=10, 4=11, 5=14 thousand steps
Each small grid square = 2 thousand steps
Both lines are drawn on the same axes

Unknowns

The day at which the two values differ the least
That smallest difference in thousand steps

Constraints

Compare the two lines only at the marked day values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Jin minus Rae) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Rae's value from Jin's at each category: day 1: 6-3=3; day 2: 9-7=2; day 3: 11-10=1; day 4: 14-11=3; day 5: 16-14=2.

3,\ 2,\ 1,\ 3,\ 2

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 3, 2, 1, 3, 2 the smallest is 1, which happens at day 3. On the graph this is where the two lines are closest.

\min(3,2,1,3,2) = 1\ \text{at day } 3

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At day 3, difference = 1 thousand steps

4 · Reviewdoes it hold up?

At day 3 the gap is 1 thousand steps, and every other category gives a larger gap, so 3 is correctly the closest. Each square is 2 thousand steps.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 3 medium

The line graph shows Town A's and Town B's temperatures, recorded over time. Find when the difference between their two values is smallest, and find that difference in degrees.

(Figure) A line graph titled "Temperatures" with both data sets drawn on the same axes. The horizontal axis is day ( $1$ , $2$ , $3$ , $4$ , $5$ ). The vertical axis is in degrees, with major gridlines at $10$ , $15$ , $20$ and each small grid square worth $1$ degrees (the lower part of the axis is cut off with a wavy line). Town A's values are $12$ , $15$ , $18$ , $20$ , $22$ degrees; Town B's values are $10$ , $14$ , $15$ , $18$ , $19$ degrees.

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1 · Understandwhat's really being asked

One line graph shows Town A's and Town B's temperatures at day 1, 2, 3, 4, 5. Town A: 12, 15, 18, 20, 22 degrees; Town B: 10, 14, 15, 18, 19 degrees (each small square = 1 degrees). I must find the day where the two values are closest and give that smallest difference.

Givens

Town A's values: 1=12, 2=15, 3=18, 4=20, 5=22 degrees
Town B's values: 1=10, 2=14, 3=15, 4=18, 5=19 degrees
Each small grid square = 1 degrees
Both lines are drawn on the same axes

Unknowns

The day at which the two values differ the least
That smallest difference in degrees

Constraints

Compare the two lines only at the marked day values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Town A minus Town B) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Town B's value from Town A's at each category: day 1: 12-10=2; day 2: 15-14=1; day 3: 18-15=3; day 4: 20-18=2; day 5: 22-19=3.

2,\ 1,\ 3,\ 2,\ 3

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 2, 1, 3, 2, 3 the smallest is 1, which happens at day 2. On the graph this is where the two lines are closest.

\min(2,1,3,2,3) = 1\ \text{at day } 2

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At day 2, difference = 1 degrees

4 · Reviewdoes it hold up?

At day 2 the gap is 1 degrees, and every other category gives a larger gap, so 2 is correctly the closest. Each square is 1 degrees.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 4 medium

The line graph shows Pot A's and Pot B's plant heights, recorded over time. Find when the difference between their two values is smallest, and find that difference in cm.

(Figure) A line graph titled "Plant Heights" with both data sets drawn on the same axes. The horizontal axis is week ( $1$ , $2$ , $3$ , $4$ , $5$ ). The vertical axis is in cm, with major gridlines at $0$ , $15$ , $30$ and each small grid square worth $3$ cm (the lower part of the axis is cut off with a wavy line). Pot A's values are $8$ , $14$ , $19$ , $23$ , $30$ cm; Pot B's values are $5$ , $12$ , $17$ , $22$ , $25$ cm.

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1 · Understandwhat's really being asked

One line graph shows Pot A's and Pot B's plant heights at week 1, 2, 3, 4, 5. Pot A: 8, 14, 19, 23, 30 cm; Pot B: 5, 12, 17, 22, 25 cm (each small square = 3 cm). I must find the week where the two values are closest and give that smallest difference.

Givens

Pot A's values: 1=8, 2=14, 3=19, 4=23, 5=30 cm
Pot B's values: 1=5, 2=12, 3=17, 4=22, 5=25 cm
Each small grid square = 3 cm
Both lines are drawn on the same axes

Unknowns

The week at which the two values differ the least
That smallest difference in cm

Constraints

Compare the two lines only at the marked week values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Pot A minus Pot B) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Pot B's value from Pot A's at each category: week 1: 8-5=3; week 2: 14-12=2; week 3: 19-17=2; week 4: 23-22=1; week 5: 30-25=5.

3,\ 2,\ 2,\ 1,\ 5

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 3, 2, 2, 1, 5 the smallest is 1, which happens at week 4. On the graph this is where the two lines are closest.

\min(3,2,2,1,5) = 1\ \text{at week } 4

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At week 4, difference = 1 cm

4 · Reviewdoes it hold up?

At week 4 the gap is 1 cm, and every other category gives a larger gap, so 4 is correctly the closest. Each square is 3 cm.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 5 medium

The line graph shows Mia's and Liam's weights, recorded over time. Find when the difference between their two values is smallest, and find that difference in kg.

(Figure) A line graph titled "Weights" with both data sets drawn on the same axes. The horizontal axis is age ( $7$ , $8$ , $9$ , $10$ , $11$ ). The vertical axis is in kg, with major gridlines at $20$ , $25$ , $30$ and each small grid square worth $1$ kg (the lower part of the axis is cut off with a wavy line). Mia's values are $24$ , $26$ , $27$ , $32$ , $30$ kg; Liam's values are $20$ , $22$ , $25$ , $26$ , $27$ kg.

Your answer (check it yourself)

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1 · Understandwhat's really being asked

One line graph shows Mia's and Liam's weights at age 7, 8, 9, 10, 11. Mia: 24, 26, 27, 32, 30 kg; Liam: 20, 22, 25, 26, 27 kg (each small square = 1 kg). I must find the age where the two values are closest and give that smallest difference.

Givens

Mia's values: 7=24, 8=26, 9=27, 10=32, 11=30 kg
Liam's values: 7=20, 8=22, 9=25, 10=26, 11=27 kg
Each small grid square = 1 kg
Both lines are drawn on the same axes

Unknowns

The age at which the two values differ the least
That smallest difference in kg

Constraints

Compare the two lines only at the marked age values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Mia minus Liam) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Liam's value from Mia's at each category: age 7: 24-20=4; age 8: 26-22=4; age 9: 27-25=2; age 10: 32-26=6; age 11: 30-27=3.

4,\ 4,\ 2,\ 6,\ 3

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 4, 4, 2, 6, 3 the smallest is 2, which happens at age 9. On the graph this is where the two lines are closest.

\min(4,4,2,6,3) = 2\ \text{at age } 9

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At age 9, difference = 2 kg

4 · Reviewdoes it hold up?

At age 9 the gap is 2 kg, and every other category gives a larger gap, so 9 is correctly the closest. Each square is 1 kg.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 6 medium

The line graph shows Sam's and Kim's weights, recorded over time. Find when the difference between their two values is smallest, and find that difference in kg.

(Figure) A line graph titled "Weights" with both data sets drawn on the same axes. The horizontal axis is age ( $8$ , $9$ , $10$ , $11$ , $12$ ). The vertical axis is in kg, with major gridlines at $20$ , $30$ , $40$ and each small grid square worth $2$ kg (the lower part of the axis is cut off with a wavy line). Sam's values are $26$ , $28$ , $31$ , $33$ , $38$ kg; Kim's values are $22$ , $27$ , $28$ , $30$ , $32$ kg.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

One line graph shows Sam's and Kim's weights at age 8, 9, 10, 11, 12. Sam: 26, 28, 31, 33, 38 kg; Kim: 22, 27, 28, 30, 32 kg (each small square = 2 kg). I must find the age where the two values are closest and give that smallest difference.

Givens

Sam's values: 8=26, 9=28, 10=31, 11=33, 12=38 kg
Kim's values: 8=22, 9=27, 10=28, 11=30, 12=32 kg
Each small grid square = 2 kg
Both lines are drawn on the same axes

Unknowns

The age at which the two values differ the least
That smallest difference in kg

Constraints

Compare the two lines only at the marked age values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Sam minus Kim) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Kim's value from Sam's at each category: age 8: 26-22=4; age 9: 28-27=1; age 10: 31-28=3; age 11: 33-30=3; age 12: 38-32=6.

4,\ 1,\ 3,\ 3,\ 6

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 4, 1, 3, 3, 6 the smallest is 1, which happens at age 9. On the graph this is where the two lines are closest.

\min(4,1,3,3,6) = 1\ \text{at age } 9

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At age 9, difference = 1 kg

4 · Reviewdoes it hold up?

At age 9 the gap is 1 kg, and every other category gives a larger gap, so 9 is correctly the closest. Each square is 2 kg.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 7 hard

The line graph shows Emma's and Leo's savings, recorded over time. Find when the difference between their two values is smallest, and find that difference in dollars.

(Figure) A line graph titled "Savings" with both data sets drawn on the same axes. The horizontal axis is week ( $1$ , $2$ , $3$ , $4$ , $5$ ). The vertical axis is in dollars, with major gridlines at $20$ , $50$ , $80$ and each small grid square worth $2$ dollars (the lower part of the axis is cut off with a wavy line). Emma's values are $30$ , $45$ , $55$ , $70$ , $80$ dollars; Leo's values are $25$ , $40$ , $52$ , $60$ , $68$ dollars.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

One line graph shows Emma's and Leo's savings at week 1, 2, 3, 4, 5. Emma: 30, 45, 55, 70, 80 dollars; Leo: 25, 40, 52, 60, 68 dollars (each small square = 2 dollars). I must find the week where the two values are closest and give that smallest difference.

Givens

Emma's values: 1=30, 2=45, 3=55, 4=70, 5=80 dollars
Leo's values: 1=25, 2=40, 3=52, 4=60, 5=68 dollars
Each small grid square = 2 dollars
Both lines are drawn on the same axes

Unknowns

The week at which the two values differ the least
That smallest difference in dollars

Constraints

Compare the two lines only at the marked week values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Emma minus Leo) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Leo's value from Emma's at each category: week 1: 30-25=5; week 2: 45-40=5; week 3: 55-52=3; week 4: 70-60=10; week 5: 80-68=12.

5,\ 5,\ 3,\ 10,\ 12

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 5, 5, 3, 10, 12 the smallest is 3, which happens at week 3. On the graph this is where the two lines are closest.

\min(5,5,3,10,12) = 3\ \text{at week } 3

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At week 3, difference = 3 dollars

4 · Reviewdoes it hold up?

At week 3 the gap is 3 dollars, and every other category gives a larger gap, so 3 is correctly the closest. Each square is 2 dollars.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Variant 8 hard

The line graph shows Ava's and Noah's heights, recorded over time. Find when the difference between their two values is smallest, and find that difference in cm.

(Figure) A line graph titled "Heights" with both data sets drawn on the same axes. The horizontal axis is age ( $6$ , $7$ , $8$ , $9$ , $10$ ). The vertical axis is in cm, with major gridlines at $100$ , $120$ , $140$ and each small grid square worth $4$ cm (the lower part of the axis is cut off with a wavy line). Ava's values are $110$ , $118$ , $124$ , $130$ , $138$ cm; Noah's values are $105$ , $116$ , $119$ , $127$ , $130$ cm.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

One line graph shows Ava's and Noah's heights at age 6, 7, 8, 9, 10. Ava: 110, 118, 124, 130, 138 cm; Noah: 105, 116, 119, 127, 130 cm (each small square = 4 cm). I must find the age where the two values are closest and give that smallest difference.

Givens

Ava's values: 6=110, 7=118, 8=124, 9=130, 10=138 cm
Noah's values: 6=105, 7=116, 8=119, 9=127, 10=130 cm
Each small grid square = 4 cm
Both lines are drawn on the same axes

Unknowns

The age at which the two values differ the least
That smallest difference in cm

Constraints

Compare the two lines only at the marked age values

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

There are only 5 categories, so I list the vertical gap (Ava minus Noah) at each and pick the smallest. On the graph the smallest gap is where the two lines come closest together vertically.

3 · Execute2 carry out the plan

1Find the gap at each category

#2 Make a Systematic List 5.MD.B.2

Subtract Noah's value from Ava's at each category: age 6: 110-105=5; age 7: 118-116=2; age 8: 124-119=5; age 9: 130-127=3; age 10: 138-130=8.

5,\ 2,\ 5,\ 3,\ 8

Listing all the differences leaves nothing out for such a small set.

2Pick the smallest gap

#1 Draw a Diagram 5.MD.B.2

Among 5, 2, 5, 3, 8 the smallest is 2, which happens at age 7. On the graph this is where the two lines are closest.

\min(5,2,5,3,8) = 2\ \text{at age } 7

The narrowest vertical gap between the two lines is the smallest difference.

Answer: At age 7, difference = 2 cm

4 · Reviewdoes it hold up?

At age 7 the gap is 2 cm, and every other category gives a larger gap, so 7 is correctly the closest. Each square is 4 cm.

Another way: Instead of subtracting, read the graph directly (tool 1): scan for where the two lines sit nearest each other vertically, then count the squares between them.

Standardsmin grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading both data sets off one graph and comparing their differences at each category

💡Takeaway. The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!

Where next?

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