Pencil Lengths to the Quarter Inch
Four students measure six pencils to the nearest quarter inch. They record the lengths as 4 1/2, 4 3/4, 5, 5 1/4, 5 1/4, and 5 1/2 inches. One student writes the last length as 5 2/4 inches and says it is a different measurement from 5 1/2. Another student says the longest pencil is 1 1/4 inches longer than the shortest. The teacher asks the group to organize the data on a line plot using a number line marked in fourths, find the difference between the longest and shortest pencils, count how many pencils are at least 5 inches long, and explain whether 5 2/4 and 5 1/2 can both describe the same length. Students must justify each answer with a model or equation.
1. What is the difference between the longest and shortest pencil? Show the subtraction.
Read the worked answer
The longest pencil is 5 1/2 inches, and the shortest is 4 1/2 inches. Write both in fourths: 5 1/2 = 5 2/4, and 4 1/2 = 4 2/4. Subtract: 5 2/4 − 4 2/4 = 1 inch. So the difference is 1 inch, not 1 1/4 inches. The second student’s claim is too large by 1/4 inch.
2. How many pencils are at least 5 inches long? Explain which data points you count.
Read the worked answer
“At least 5” means 5 or more, so include 5 and every length greater than 5. The data are 4 1/2, 4 3/4, 5, 5 1/4, 5 1/4, and 5 1/2. Count 5, 5 1/4, 5 1/4, and 5 1/2: that is 4 pencils. Do not count 4 1/2 or 4 3/4 because they are less than 5.
3. Can 5 2/4 and 5 1/2 describe the same length? Explain using equivalent fractions or a number line.
Read the worked answer
Yes. 2/4 is equivalent to 1/2 because 2 out of 4 equal parts is the same as 1 out of 2 equal parts. So 5 2/4 = 5 1/2. On a number line marked in fourths, 5 1/2 is the same point as 5 2/4: start at 5 and move 2 fourths. Writing the same measurement with a different equivalent fraction does not make it a different length.