Reading a Line Plot with Quarter-Inch Marks
Students measure bean seedlings to the nearest quarter inch and record the lengths on a line plot. The finished plot shows five stacks of X marks above the lengths 1/2 inch, 3/4 inch, 1 inch, 1 1/4 inches, and 1 1/2 inches, containing 3, 5, 4, 2, and 2 marks. The teacher asks three questions: how many seedlings were measured in all, how many were at least 1 inch long, and how the longest seedling compares with the shortest. Two students answer the last question differently. Jada says the longest is 1 inch longer than the shortest. Kai says the longest is 3 times as long as the shortest. A visitor then claims the plot must be incomplete because it shows only about a dozen seedlings. Students must decide which statements the plot actually supports, using the marks themselves rather than the labels alone.
1. How many seedlings were measured in all, and how many were at least 1 inch long? Show how the marks give each total.
Read the worked answer
Each X is one seedling, so add the heights of all five stacks: 3 + 5 + 4 + 2 + 2 = 16 seedlings measured.
At least 1 inch means include the 1-inch stack and everything to its right: 4 (at 1 in) + 2 (at 1 1/4 in) + 2 (at 1 1/2 in) = 8 seedlings.
The two shorter stacks (3 at 1/2 in and 5 at 3/4 in) do not count, which is why 16 − 8 = 8 seedlings were shorter than 1 inch.
2. Jada and Kai disagree about the longest and shortest seedlings. Decide who is correct, and explain why both statements can be true at once.
Read the worked answer
Both are correct, because they answer different questions.
Jada answers a difference question: 1 1/2 − 1/2 = 1 inch longer.
Kai answers a times-as-many question: lay 1/2-inch lengths end to end and it takes three of them to reach 1 1/2 inches (1/2 + 1/2 + 1/2 = 1 1/2), so the longest is 3 times the shortest.
The word times asks how many copies of the shorter length fit inside the longer one; subtracting asks how far apart the two lengths are. The plot supports both, so neither student should be marked wrong - but the class should be clear about which comparison the question asked for.
3. Evaluate two claims: first, that half the seedlings are 3/4 inch or shorter; second, that more seedlings are exactly 1 inch than exactly 3/4 inch. Also respond to the visitor's dozen claim.
Read the worked answer
First claim: 3/4 inch or shorter means the 1/2-inch and 3/4-inch stacks, giving 3 + 5 = 8 seedlings. Since 8 is half of 16, the claim is correct.
Second claim: the 1-inch stack has 4 marks and the 3/4-inch stack has 5 marks, and 4 is not more than 5, so that claim is false.
Visitor's claim: a dozen is 12, but the plot holds 16 marks, so the plot is being undercounted. A reasonable reply is that the marks total 16, and 16 is not about 12 - the mistake comes from glancing at five stacks and guessing rather than adding heights.