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← 4-2 · Largest vertical gap between two line graphs · Read and Scale a Data Graph

Largest vertical gap between two line graphs · 8 practice problems

5.MD.B.2

Generated variants — 8

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

Variant 1 answer: At week 3, difference = 3 dollars

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 (11, 22, 33, 44, 55). The vertical axis is in dollars, with major gridlines at 2020, 5050, 8080 and each small grid square worth 22 dollars (the lower part of the axis is cut off with a wavy line). Emma's values are 3030, 4545, 5555, 7070, 8080 dollars; Leo's values are 2525, 4040, 5252, 6060, 6868 dollars.

Savings 20 50 80 (dollars) 1 2 3 4 5 Week Emma Leo
Show solution

Understand

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

Plan

#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.

Execute

#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, 125,\ 5,\ 3,\ 10,\ 12
Listing all the differences leaves nothing out for such a small set.
#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 at week 3\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

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 2 answer: At age 9, difference = 2 kg

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 (77, 88, 99, 1010, 1111). The vertical axis is in kg, with major gridlines at 2020, 2525, 3030 and each small grid square worth 11 kg (the lower part of the axis is cut off with a wavy line). Mia's values are 2424, 2626, 2727, 3232, 3030 kg; Liam's values are 2020, 2222, 2525, 2626, 2727 kg.

Weights 20 25 30 (kg) 7 8 9 10 11 Age Mia Liam
Show solution

Understand

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

Plan

#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.

Execute

#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, 34,\ 4,\ 2,\ 6,\ 3
Listing all the differences leaves nothing out for such a small set.
#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 at age 9\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

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 3 answer: At age 7, difference = 2 cm

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 (66, 77, 88, 99, 1010). The vertical axis is in cm, with major gridlines at 100100, 120120, 140140 and each small grid square worth 44 cm (the lower part of the axis is cut off with a wavy line). Ava's values are 110110, 118118, 124124, 130130, 138138 cm; Noah's values are 105105, 116116, 119119, 128128, 130130 cm.

Heights 100 120 140 (cm) 6 7 8 9 10 Age Ava Noah
Show solution

Understand

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, 128, 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=128, 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

Plan

#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.

Execute

#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: 130-128=2; age 10: 138-130=8.
5, 2, 5, 2, 85,\ 2,\ 5,\ 2,\ 8
Listing all the differences leaves nothing out for such a small set.
#1 Draw a Diagram 5.MD.B.2
Among 5, 2, 5, 2, 8 the smallest is 2, which happens at age 7. On the graph this is where the two lines are closest.
min⁡(5,2,5,2,8)=2 at age 7\min(5,2,5,2,8) = 2\ \text{at age } 7
The narrowest vertical gap between the two lines is the smallest difference.
Answer: At age 7, difference = 2 cm

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 4 answer: At day 2, difference = 1 degrees

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 (11, 22, 33, 44, 55). The vertical axis is in degrees, with major gridlines at 1010, 1515, 2020 and each small grid square worth 11 degrees (the lower part of the axis is cut off with a wavy line). Town A's values are 1212, 1515, 1818, 2020, 2222 degrees; Town B's values are 1010, 1414, 1515, 1919, 2121 degrees.

Temperatures 10 15 20 (degrees) 1 2 3 4 5 Day Town A Town B
Show solution

Understand

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, 19, 21 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=19, 5=21 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

Plan

#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.

Execute

#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: 20-19=1; day 5: 22-21=1.
2, 1, 3, 1, 12,\ 1,\ 3,\ 1,\ 1
Listing all the differences leaves nothing out for such a small set.
#1 Draw a Diagram 5.MD.B.2
Among 2, 1, 3, 1, 1 the smallest is 1, which happens at day 2. On the graph this is where the two lines are closest.
min⁡(2,1,3,1,1)=1 at day 2\min(2,1,3,1,1) = 1\ \text{at day } 2
The narrowest vertical gap between the two lines is the smallest difference.
Answer: At day 2, difference = 1 degrees

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 5 answer: At day 1, difference = 1 thousand steps

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 (11, 22, 33, 44, 55). The vertical axis is in thousand steps, with major gridlines at 00, 88, 1616 and each small grid square worth 22 thousand steps (the lower part of the axis is cut off with a wavy line). Jin's values are 66, 99, 1111, 1414, 1616 thousand steps; Rae's values are 55, 88, 1010, 1111, 1515 thousand steps.

Steps Walked 0 8 16 (thousand steps) 1 2 3 4 5 Day Jin Rae
Show solution

Understand

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: 5, 8, 10, 11, 15 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=5, 2=8, 3=10, 4=11, 5=15 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

Plan

#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.

Execute

#2 Make a Systematic List 5.MD.B.2
Subtract Rae's value from Jin's at each category: day 1: 6-5=1; day 2: 9-8=1; day 3: 11-10=1; day 4: 14-11=3; day 5: 16-15=1.
1, 1, 1, 3, 11,\ 1,\ 1,\ 3,\ 1
Listing all the differences leaves nothing out for such a small set.
#1 Draw a Diagram 5.MD.B.2
Among 1, 1, 1, 3, 1 the smallest is 1, which happens at day 1. On the graph this is where the two lines are closest.
min⁡(1,1,1,3,1)=1 at day 1\min(1,1,1,3,1) = 1\ \text{at day } 1
The narrowest vertical gap between the two lines is the smallest difference.
Answer: At day 1, difference = 1 thousand steps

Review

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

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 6 answer: At age 9, difference = 1 kg

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 (88, 99, 1010, 1111, 1212). The vertical axis is in kg, with major gridlines at 2020, 3030, 4040 and each small grid square worth 22 kg (the lower part of the axis is cut off with a wavy line). Sam's values are 2626, 2828, 3131, 3333, 3838 kg; Kim's values are 2222, 2727, 2828, 3030, 3232 kg.

Weights 20 30 40 (kg) 8 9 10 11 12 Age Sam Kim
Show solution

Understand

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

Plan

#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.

Execute

#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, 64,\ 1,\ 3,\ 3,\ 6
Listing all the differences leaves nothing out for such a small set.
#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 at age 9\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

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 7 answer: At week 4, difference = 1 cm

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 (11, 22, 33, 44, 55). The vertical axis is in cm, with major gridlines at 00, 1515, 3030 and each small grid square worth 33 cm (the lower part of the axis is cut off with a wavy line). Pot A's values are 88, 1414, 1919, 2323, 3030 cm; Pot B's values are 55, 1212, 1717, 2222, 2525 cm.

Plant Heights 0 15 30 (cm) 1 2 3 4 5 Week Pot A Pot B
Show solution

Understand

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

Plan

#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.

Execute

#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, 53,\ 2,\ 2,\ 1,\ 5
Listing all the differences leaves nothing out for such a small set.
#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 at week 4\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

Review

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.

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.

Standards · min 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
💡 The two lines are closest where their gap is smallest - just check each point and pick the tiniest difference!
Variant 8 answer: At month 1, difference = 1 books

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 (11, 22, 33, 44, 55). The vertical axis is in books, with major gridlines at 00, 55, 1010 and each small grid square worth 11 books (the lower part of the axis is cut off with a wavy line). Sofia's values are 44, 77, 99, 1212, 1414 books; Max's values are 33, 66, 88, 99, 1313 books.

Reading Counts 0 5 10 (books) 1 2 3 4 5 Month Sofia Max
Show solution

Understand

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: 3, 6, 8, 9, 13 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=3, 2=6, 3=8, 4=9, 5=13 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

Plan

#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.

Execute

#2 Make a Systematic List 5.MD.B.2
Subtract Max's value from Sofia's at each category: month 1: 4-3=1; month 2: 7-6=1; month 3: 9-8=1; month 4: 12-9=3; month 5: 14-13=1.
1, 1, 1, 3, 11,\ 1,\ 1,\ 3,\ 1
Listing all the differences leaves nothing out for such a small set.
#1 Draw a Diagram 5.MD.B.2
Among 1, 1, 1, 3, 1 the smallest is 1, which happens at month 1. On the graph this is where the two lines are closest.
min⁡(1,1,1,3,1)=1 at month 1\min(1,1,1,3,1) = 1\ \text{at month } 1
The narrowest vertical gap between the two lines is the smallest difference.
Answer: At month 1, difference = 1 books

Review

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

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.

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