Largest vertical gap between two line graphs

5.MD.B.2 · adapt · grade 5

Archetype: Read and Scale a Data Graph · step in a 21-type progression

Representative Problem

The line graph shows Mia's and Liam's body weights, recorded each May. Find when the difference between their two weights is smallest, and find that difference in kilograms.

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

(Note: weights are kept in kilograms because the answer is read directly off this metric line graph; the figure has not been redrawn in imperial units.)

Show solution

Understand

One line graph shows Mia's and Liam's weights at ages 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 weights are closest and give that smallest difference.

Givens

Mia's weights: age7 = 24, age8 = 26, age9 = 27, age10 = 32, age11 = 30 kg
Liam's weights: age7 = 20, age8 = 22, age9 = 25, age10 = 26, age11 = 27 kg
Each small grid square = 1 kg
Both lines are drawn on the same axes

Unknowns

The age at which the two weights differ the least
That smallest difference in kilograms

Constraints

Compare the two lines only at the marked ages

Plan

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

There are only five ages, so I list the vertical gap (Mia minus Liam) at each age and pick the smallest. Looking at 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 weight from Mia's at each age: age7: 24-20=4, age8: 26-22=4, age9: 27-25=2, age10: 32-26=6, age11: 30-27=3.

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

Listing all five 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, only one square apart times 2.

\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 Mia is 27 and Liam is 25, a 2 kg gap, and every other age gives 3 kg or more, so age 9 is correctly the closest. Each square is 1 kg, and the lines there are 2 squares apart, matching the answer.

Instead of subtracting, read the graph directly (tool 1): scan for where the red and blue dots sit nearest each other vertically - that is age 9 - then count 2 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 age

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