Line Plots in Halves and Fourths

One dot for each measurement

A class grows nine seedlings and measures each one to the nearest quarter of an inch. To show all nine lengths at once, draw a number line from 1 inch to 3 inches and mark it in quarters: 1, 1¼, 1½, 1¾, 2, and on up to 3. Then put one dot above the mark for each seedling. This drawing is called a line plot.

The shortest seedling measured 1½ inches, so one dot goes above 1½. The next two both measured 1¾ inches. The first of them goes just above the line at 1¾, and the second goes on top of it.

Nine dots for nine seedlings. Each dot sits above the length that one seedling measured.

The tallest column

When two seedlings have the same length, their dots stack up in one column. So the height of a column tells you how many seedlings had that length.

The tallest column is above 2, with three dots: three seedlings measured exactly 2 inches, more than any other length. The column at 1¾ has two dots, and every other length has one. Count all the dots to check the plot: 1 + 2 + 3 + 1 + 1 + 1 = 9, one for each seedling.

The column above 2 is the tallest: 2 inches is the length measured most often.

Why the scale needs quarters

The seedlings were measured to the nearest quarter inch, so the scale needs a mark for every quarter. A scale marked only in halves has five marks from 1 to 3: 1, 1½, 2, 2½ and 3.

On that scale the dots at 1½, 2 and 2½ still sit on a mark. But the dots at 1¾, 2¼ and 2¾ land between two marks, and no number on the scale names them. The scale has to be marked as finely as the measurements were made.

The same nine seedlings on a scale of halves. The dots at 1¾, 2¼ and 2¾ fall between the marks.

The shortest and the tallest

The shortest seedling is the dot farthest to the left, at 1½ inches. The tallest is the dot farthest to the right, at 2¾ inches. To find how much taller the tallest one is, take the shortest length away from the tallest: 2¾ − 1½.

Count along the scale. From 1½ to 2½ is 1 whole inch, and from 2½ to 2¾ is one more quarter. So 2¾ − 1½ = 1¼ inches.

This is a difference, so it is found by taking away. Adding the two lengths, or giving the tallest length on its own, does not say how much taller one seedling is than the other.

From 1½ to 2½ is 1 inch, and a quarter more reaches 2¾: the tallest is 1¼ inches taller.

Adding a column

The two dots at 1¾ are two seedlings, each 1¾ inches long. Laid end to end, they come to 1¾ + 1¾.

Add the whole inches first: 1 + 1 = 2. Then add the quarters: $\frac{3}{4}+\frac{3}{4}=\frac{6}{4}$¾ + ¾ = 6/4. Four quarters make a whole, so $\frac{6}{4}$6/4 is 1 whole and $\frac{2}{4}$2/4 more, which is 1½. Together that is 2 + 1½ = 3½ inches.

Each dot is one seedling, so a column of two dots uses its length twice. The answer is not 1¾, which is only one of them, and it is not 2¾, which adds 1 inch in place of the second seedling.

Two hops of 1¾ inches, one for each seedling, end at 3½ inches.