Reading-Time Line Plot: Average, Threshold, and Misread Graphs
Each of the 20 students in Room 12 logged Tuesday reading time to the nearest quarter hour, and the class built a line plot: 2 students read 1/2 hour, 4 read 3/4 hour, 6 read 1 hour, 5 read 1 1/4 hours, and 3 read 1 1/2 hours. The principal has promised a celebration if the class average is at least 1 hour per student and if more than half the class read at least 1 hour. Two students read the line plot differently. Omar says, “The tallest stack is at 1 hour, so the average is 1 hour,” and Leah says, “Only 6 students met the goal, because only 6 read exactly 1 hour.” The teacher must settle both readings and decide whether to announce the celebration at 2:15, with no time to rebuild the graph.
1. Compute the class's total reading time, first in hours and then in minutes, showing how the line plot data enter the calculation.
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Multiply each value by its frequency: 2 × 1/2 = 1 hour; 4 × 3/4 = 3 hours; 6 × 1 = 6 hours; 5 × 1 1/4 = 6 1/4 hours; 3 × 1 1/2 = 4 1/2 hours. Total: 1 + 3 + 6 + 6.25 + 4.5 = 20.75, or 20 3/4 hours. Convert by multiplying by 60: 20 × 60 = 1,200 minutes and 3/4 × 60 = 45 minutes, so 1,245 minutes. Checking in minutes directly: 2 × 30 = 60, 4 × 45 = 180, 6 × 60 = 360, 5 × 75 = 375, 3 × 90 = 270, and 60 + 180 + 360 + 375 + 270 = 1,245.
2. Decide whether the class earned the celebration, using both conditions. Show the comparison rather than computing a long division.
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Condition 1: an average of 1 hour for 20 students requires 20 × 1 = 20 hours total. The actual total is 20.75 hours, which is greater than 20, so the average is above 1 hour - about 1 hour and 2 minutes per student. Condition 2: “at least 1 hour” includes 1 hour, 1 1/4 hours, and 1 1/2 hours, so the count is 6 + 5 + 3 = 14 students. Since 14 out of 20 is more than half (10), the second condition is also met. Both conditions hold, so the teacher can announce the celebration at 2:15 with the totals as evidence.
3. Diagnose Omar's and Leah's reasoning about the line plot. What does each student appear to misunderstand?
Read the worked answer
Omar is reading the tallest stack as the average. The tallest stack shows the mode, the most common value, which here is 1 hour; an average depends on every value, and the 8 students who read more than 1 hour pull it above the mode. Leah is reading the goal as “exactly 1 hour” when the principal said “at least 1 hour,” which includes the 5 students at 1 1/4 hours and the 3 at 1 1/2 hours, giving 14 rather than 6. Both errors are graph-interpretation errors rather than arithmetic errors, so the reteach is a two-question routine for any line plot: what does each stack contribute to the total, and does the wording say exactly or at least?