Grade 5 · 3.MD.B.4 · 4.MD.B.4

Line plots with fractions

Ten seashells are measured to the nearest quarter inch. How can every length be shown in one picture, and what can that picture tell you?

You need

Fractions on the number lineGrade 3 Making a dot plotGrade 2

Where you are 3 in Dot plots

Statistics7

Data displays12

Dot plots3

Making a dot plot Reading dot plots: clusters, gaps and peaks

Line plots with fractions

When you reach for it

On a nature walk, Akua picks up seashells and Kagiso picks up pinecones. In science class, Laleh’s class measures ladybugs. Every one of these things has a length, and almost none of the lengths is a whole number of inches. A shell might be \displaystyle 1\frac{1}{2} inches long, a ladybug only \displaystyle \frac{3}{8} of an inch.

A list of ten or twelve lengths is hard to take in. You cannot see at a glance which length came up most, or how far apart the shortest and the longest are. A line plot shows all of them at once. It is a dot plot: a number line with one dot stacked above a value each time that value was measured. The only new thing here is that the number line is marked in fractions: halves, quarters or eighths of an inch.

Reach for a line plot with fractions when you have many measurements of the same kind, made to the nearest half, quarter or eighth of a unit, and you want to see them together, answer questions about them, or share their total out evenly.

The lesson takes three days. Part 1 measures to a fraction of an inch and makes the plot. Part 2 reads the plot and works with the numbers on it. Part 3 uses the plot to share a total out equally.

The procedure

Part 1: measuring and plotting

Measuring to a fraction of an inch. A ruler cuts each inch into equal parts. On most rulers the inch is cut into 8 equal parts, so the smallest step is \displaystyle \frac{1}{8} inch. The ticks have different heights to help you count: the inch tick is the longest, the half inch tick is next, then the quarter inch ticks, and the eighth inch ticks are the shortest (Measuring to the half and quarter inch).

Figure 1The shell ends just past \displaystyle 1\frac{3}{8} inches. To the nearest eighth it is \displaystyle 1\frac{3}{8} inches; to the nearest quarter, and to the nearest half, it is \displaystyle 1\frac{1}{2} inches.

Line up one end of the object with 0 and look at where the other end falls. “To the nearest eighth” means: of all the eighth inch ticks, pick the one closest to the end. The shell above ends a little past the \displaystyle 1\frac{3}{8} tick, and that tick is the closest eighth, so to the nearest eighth inch the shell is \displaystyle 1\frac{3}{8} inches long.

“To the nearest quarter” uses only the quarter inch ticks: 1, \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2}, \displaystyle 1\frac{3}{4}, 2. The end lies between \displaystyle 1\frac{1}{4} and \displaystyle 1\frac{1}{2}, and it is much closer to \displaystyle 1\frac{1}{2}, so to the nearest quarter inch the shell is \displaystyle 1\frac{1}{2} inches. To the nearest half inch it is also \displaystyle 1\frac{1}{2} inches. The same shell can be reported three ways, so every measurement in a set must be made to the same nearest mark, or they cannot be compared fairly.

Making the line plot. A line plot is a number line, and a number line must have equal distances for equal amounts (Fractions on the number line). So the line is marked every \displaystyle \frac{1}{4} inch for quarter inch data, every \displaystyle \frac{1}{8} inch for eighth inch data, with the marks evenly spaced and none skipped, even where no measurement landed. Skipping a mark would make one step look the same size as two.

Start the line at or a little below the shortest measurement and end it at or a little above the longest, so every value has a mark. Label the marks, and write under the line what was measured and in what unit, such as “seashell length (inches)”.

Then put one dot above the matching mark for each measurement. Two shells of the same length get two dots, stacked one above the other. The number of dots must equal the number of things measured: that is the check that none was missed or counted twice.

Two names for the same number sit at the same mark. If one shell was written down as \displaystyle 1\frac{2}{4} inches and another as \displaystyle 1\frac{1}{2} inches, they are the same length, because \displaystyle \frac{2}{4} = \frac{1}{2} (Equivalent fractions), and both dots go above the \displaystyle 1\frac{1}{2} mark.

Part 2: reading the plot

Every question about the data is a question about dots and marks.

How many were measured at one length: count the dots above that mark.
The most common length is the mark with the tallest stack.
The shortest and longest are the leftmost and rightmost dots. The difference between them, longest minus shortest, says how spread out the lengths are. It is found by subtracting fractions or mixed numbers (Subtracting mixed numbers).
The total length at one mark is the value added once for every dot. Three dots above \displaystyle 2\frac{1}{2} stand for \displaystyle 2\frac{1}{2} + 2\frac{1}{2} + 2\frac{1}{2}, which is \displaystyle 3 \times 2\frac{1}{2} = 7\frac{1}{2} (A fraction times a whole number).
The total of everything is the sum of the totals at each mark. When the marks have different denominators, such as \displaystyle \frac{1}{4} and \displaystyle \frac{3}{8}, rename to the same size pieces before adding (Adding and subtracting fractions with unlike denominators, Adding mixed numbers).

The plot does not do the arithmetic for you. What it does is organize it: each stack becomes one multiplication, so twelve separate additions turn into four or five.

Part 3: sharing the total equally

Here is a new question. Suppose all the water in several beakers were poured together and then poured back out so that every beaker held the same amount. How much would each beaker hold?

Pouring the water together gives the total, which Part 2 showed how to find. Pouring it back out equally shares that total among the beakers. So the equal share is

\text{equal share} = \text{total} \div \text{number of measurements}

When the total is a fraction, this is a fraction divided by a whole number, which is the sharing meaning of division (Dividing fractions: sharing into equal parts). When the total is a whole number of units, it is A fraction as division: \displaystyle 4 \div 12 = \frac{4}{12}.

You can also picture the sharing on the plot itself. Every dot above the equal share has extra to give away; every dot below it is short. Pour from the full ones into the short ones until they are all level. The amount given away must equal the amount taken in, because no water is lost or added. That gives a check on the answer.

The equal share always lies between the shortest and the longest value. It does not have to be a value on the plot, or even a mark on the line. In later work this number gets a name, the mean, and Mean as fair share: computing the mean studies it on its own.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3, 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

Akua’s seashells on a quarter inch line

Akua measures 10 seashells to the nearest quarter inch. Her lengths, in inches, are \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2}, \displaystyle 1\frac{3}{4}, \displaystyle 1\frac{1}{2}, 2, \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2}, \displaystyle 2\frac{1}{4}, \displaystyle 1\frac{3}{4}, \displaystyle 1\frac{1}{2}. Make a line plot.

Solution

The lengths were measured to the nearest quarter inch, so the line is marked every \displaystyle \frac{1}{4} inch. The shortest length is \displaystyle 1\frac{1}{4} and the longest is \displaystyle 2\frac{1}{4}, so a line from 1 to \displaystyle 2\frac{1}{2} holds them all with a little room at each end. Its marks are 1, \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2}, \displaystyle 1\frac{3}{4}, 2, \displaystyle 2\frac{1}{4} and \displaystyle 2\frac{1}{2}, evenly spaced.

Now go down the list and put one dot above the mark for each shell. Counting them up:

length (inches)	\displaystyle 1\frac{1}{4}	\displaystyle 1\frac{1}{2}	\displaystyle 1\frac{3}{4}	2	\displaystyle 2\frac{1}{4}
dots	2	4	2	1	1

Figure 2Akua’s 10 seashells, one dot for each shell.

Check: 2 + 4 + 2 + 1 + 1 = 10 dots for 10 shells, so none was missed.

Example 2

Ladybugs written down in different names

Laleh’s class measures 12 ladybugs to the nearest eighth of an inch. Different students wrote the lengths in different ways. In inches: \displaystyle \frac{3}{8}, \displaystyle \frac{1}{4}, \displaystyle \frac{1}{2}, \displaystyle \frac{3}{8}, \displaystyle \frac{2}{8}, \displaystyle \frac{3}{8}, \displaystyle \frac{1}{8}, \displaystyle \frac{4}{8}, \displaystyle \frac{3}{8}, \displaystyle \frac{1}{4}, \displaystyle \frac{3}{8}, \displaystyle \frac{1}{4}. Make a line plot.

Solution

The measuring was to the nearest eighth, so the line is marked every \displaystyle \frac{1}{8} inch. To see which mark each length belongs to, write every length in eighths. Multiplying the top and bottom of a fraction by the same number gives an equal fraction:

\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}

\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}

So \displaystyle \frac{1}{4} and \displaystyle \frac{2}{8} share one mark, and \displaystyle \frac{1}{2} and \displaystyle \frac{4}{8} share another. In eighths the list is 3, 2, 4, 3, 2, 3, 1, 4, 3, 2, 3, 2. Counting:

length (inches)	\displaystyle \frac{1}{8}	\displaystyle \frac{2}{8}	\displaystyle \frac{3}{8}	\displaystyle \frac{4}{8}
dots	1	4	5	2

A line from 0 to \displaystyle \frac{5}{8} holds them all. On the plot the marks are labeled in lowest terms, so the \displaystyle \frac{2}{8} mark reads \displaystyle \frac{1}{4} and the \displaystyle \frac{4}{8} mark reads \displaystyle \frac{1}{2}. Either way of labeling is correct.

Figure 3The class’s 12 ladybugs. The marks are labeled in lowest terms, so \displaystyle \frac{2}{8} is written \displaystyle \frac{1}{4} and \displaystyle \frac{4}{8} is written \displaystyle \frac{1}{2}.

Check: 1 + 4 + 5 + 2 = 12 dots for 12 ladybugs.

Example 3

Reading Kagiso’s pinecone plot

Kagiso measured his pinecones to the nearest eighth of an inch and made the line plot below.

Figure 4Kagiso’s 11 pinecones, measured to the nearest eighth of an inch.

How many pinecones did he measure? What is the most common length? What is the difference between the longest and the shortest? How many pinecones are shorter than \displaystyle 2\frac{1}{2} inches?

Solution

Each dot is one pinecone, so count all the dots: 1 + 2 + 3 + 2 + 1 + 2 = 11 pinecones.

The tallest stack has 3 dots, above \displaystyle 2\frac{1}{2}. The most common length is \displaystyle 2\frac{1}{2} inches.

The leftmost dot is at \displaystyle 2\frac{1}{8}, so the shortest pinecone is \displaystyle 2\frac{1}{8} inches. The rightmost dots are at 3, so the longest is 3 inches. The difference is

3 - 2\frac{1}{8} = \frac{7}{8}

because \displaystyle 2\frac{1}{8} is \displaystyle \frac{7}{8} short of 3: counting up \displaystyle \frac{7}{8} from \displaystyle 2\frac{1}{8} lands on \displaystyle 2\frac{8}{8} = 3. The longest pinecone is \displaystyle \frac{7}{8} inch longer than the shortest.

The pinecones shorter than \displaystyle 2\frac{1}{2} inches are the dots to the left of that mark: 1 at \displaystyle 2\frac{1}{8} and 2 at \displaystyle 2\frac{3}{8}, so 3 pinecones. The dots above \displaystyle 2\frac{1}{2} itself are not shorter, so they are not counted.

Example 4

Total length of the long pinecones

Kagiso lines up every pinecone from the plot in Example 3 that is longer than \displaystyle 2\frac{1}{2} inches, end to end. How long is the row?

Solution

The pinecones longer than \displaystyle 2\frac{1}{2} inches are the dots to the right of that mark: 2 at \displaystyle 2\frac{5}{8}, 1 at \displaystyle 2\frac{3}{4} and 2 at 3. Find the total at each mark first. Two pinecones of \displaystyle 2\frac{5}{8} inches:

2 \times 2\frac{5}{8} = 4\frac{10}{8} = 5\frac{2}{8} = 5\frac{1}{4}

The ten eighths are one whole and two eighths more. One pinecone of \displaystyle 2\frac{3}{4} inches gives \displaystyle 2\frac{3}{4}. Two pinecones of 3 inches give 6. Now add the three totals. The fractions \displaystyle \frac{1}{4} and \displaystyle \frac{3}{4} are already in the same size pieces, so

5\frac{1}{4} + 2\frac{3}{4} = 7\frac{4}{4} = 8

and then 8 + 6 = 14. The row of 5 pinecones is 14 inches long.

Example 5

One more pinecone

On the way home, Kagiso finds one more pinecone and adds it to the plot from Example 3. Now the difference between the longest and the shortest pinecone is \displaystyle 1\frac{1}{4} inches. How long could the new pinecone be?

Solution

Before, the difference was \displaystyle \frac{7}{8} inch. It got bigger, so the new pinecone must be a new longest or a new shortest one. A pinecone in between would leave the shortest and longest as they were, and the difference would stay \displaystyle \frac{7}{8}.

If it is the new longest, it is \displaystyle 1\frac{1}{4} inches longer than the shortest, \displaystyle 2\frac{1}{8}. Rename \displaystyle 1\frac{1}{4} as \displaystyle 1\frac{2}{8} so the pieces match:

2\frac{1}{8} + 1\frac{2}{8} = 3\frac{3}{8}

If it is the new shortest, it is \displaystyle 1\frac{1}{4} inches shorter than the longest, 3:

3 - 1\frac{1}{4} = 1\frac{3}{4}

The new pinecone is either \displaystyle 3\frac{3}{8} inches or \displaystyle 1\frac{3}{4} inches long. Either way the line must be made longer to hold it: out to \displaystyle 3\frac{3}{8}, or back to \displaystyle 1\frac{3}{4}, still marked every eighth. Check: \displaystyle 3\frac{3}{8} - 2\frac{1}{8} = 1\frac{2}{8} = 1\frac{1}{4}, and \displaystyle 3 - 1\frac{3}{4} = 1\frac{1}{4}.

Example 6

Leveling the beakers

Laleh’s class keeps water in 8 beakers for the ladybug jars. The line plot shows how much water is in each beaker, in liters. If all the water were poured together and then shared equally among the 8 beakers, how much would each beaker hold?

Figure 5Water in 8 beakers, before it is shared out.

Solution

First the total. Work in eighths, so every amount is in the same size pieces: \displaystyle \frac{1}{4} = \frac{2}{8}, \displaystyle \frac{1}{2} = \frac{4}{8} and \displaystyle \frac{3}{4} = \frac{6}{8}. The totals at the marks are \displaystyle 1 \times \frac{1}{8} = \frac{1}{8}, then \displaystyle 3 \times \frac{2}{8} = \frac{6}{8}, then \displaystyle 1 \times \frac{3}{8} = \frac{3}{8}, then \displaystyle 2 \times \frac{4}{8} = \frac{8}{8}, then \displaystyle 1 \times \frac{6}{8} = \frac{6}{8}. Adding,

\begin{aligned} &\frac{1}{8} + \frac{6}{8} + \frac{3}{8} + \frac{8}{8} + \frac{6}{8} \\ &= \frac{24}{8} = 3 \end{aligned}

so there are 3 liters of water in all. Shared equally among 8 beakers, each holds

3 \div 8 = \frac{3}{8}

of a liter.

Check by leveling. The beakers above \displaystyle \frac{3}{8} give water away: each of the two \displaystyle \frac{4}{8} beakers gives \displaystyle \frac{1}{8}, and the \displaystyle \frac{6}{8} beaker gives \displaystyle \frac{3}{8}, so \displaystyle \frac{5}{8} is given in all. The beakers below \displaystyle \frac{3}{8} take water in: the \displaystyle \frac{1}{8} beaker takes \displaystyle \frac{2}{8}, and each of the three \displaystyle \frac{2}{8} beakers takes \displaystyle \frac{1}{8}, so \displaystyle \frac{5}{8} is taken in all. Given equals taken, so the water comes out exactly level at \displaystyle \frac{3}{8} liter. The beaker already at \displaystyle \frac{3}{8} neither gives nor takes.

Example 7

A share that is not on the line

Laleh’s class lines up all 12 ladybugs from Example 2, nose to tail, on a sheet of paper. How long is the line of ladybugs? If the ladybugs were all the same length, with the same total, how long would each one be?

Solution

From the plot, in eighths: 1 ladybug of \displaystyle \frac{1}{8}, 4 of \displaystyle \frac{2}{8}, 5 of \displaystyle \frac{3}{8} and 2 of \displaystyle \frac{4}{8}. The totals at the marks are \displaystyle \frac{1}{8}, \displaystyle \frac{8}{8}, \displaystyle \frac{15}{8} and \displaystyle \frac{8}{8}, and

\frac{1}{8} + \frac{8}{8} + \frac{15}{8} + \frac{8}{8} = \frac{32}{8} = 4

So the line of ladybugs is 4 inches long. Sharing 4 inches equally among 12 ladybugs is 4 \div 12, and a whole number divided by a whole number can be written as a fraction:

4 \div 12 = \frac{4}{12} = \frac{1}{3}

Each ladybug would be \displaystyle \frac{1}{3} inch long.

Notice that \displaystyle \frac{1}{3} is not an eighth: it is not one of the marks on the line. To see where it falls, write the neighboring marks in twenty-fourths: \displaystyle \frac{2}{8} = \frac{6}{24}, \displaystyle \frac{1}{3} = \frac{8}{24} and \displaystyle \frac{3}{8} = \frac{9}{24}. So \displaystyle \frac{1}{3} lies between the \displaystyle \frac{1}{4} mark and the \displaystyle \frac{3}{8} mark. No ladybug was actually \displaystyle \frac{1}{3} inch long; the equal share is a number made from all of them, not a length that one of them had. Check: \displaystyle 12 \times \frac{1}{3} = \frac{12}{3} = 4 inches.

Boundaries

The data must be measurements of one kind in one unit. Lengths of shells in inches can share a plot. A shell in inches and a pinecone in centimeters cannot, and neither can a length and an amount of water.
Every value needs a mark. A plot marked in quarters has no place for \displaystyle \frac{3}{8}. If any value is in eighths, the whole line must be marked in eighths. Every quarter mark is still there, because \displaystyle \frac{1}{4} = \frac{2}{8}, \displaystyle \frac{1}{2} = \frac{4}{8} and \displaystyle \frac{3}{4} = \frac{6}{8}.
The marks must be evenly spaced, with none skipped. Otherwise the line is not a number line, and the distances on it mean nothing.
“To the nearest” needs one closest mark. An end that falls exactly halfway between two marks has no single nearest one. Then measure again, or use the next smaller unit.
A line plot loses the order and the names. It shows how many shells had each length, not which shell was which or which was picked up first.
The equal share answers “what if it were shared equally”. It is not one of the measurements and need not be a mark. Sharing makes sense for amounts that could really be pooled, like water. For lengths of ladybugs it is a “what if”: the total, spread evenly.

Where this goes next

A line plot is the first picture of data where the values are fractions, not counts. The next step reads plots for their shape: where the dots bunch up, where there are gaps, and where the peak is (Reading dot plots: clusters, gaps and peaks). The equal share of Part 3 is the first look at the mean, the most used number in statistics; Mean as fair share: computing the mean treats it as a fair share, and Mean as balance point shows that the same number is the point where a dot plot balances. The difference between longest and shortest grows into the range, one of the ways to measure how spread out data are (Range and interquartile range).

Exercises

Exercises 1 to 7 go with Part 1, 8 to 15 with Part 2, 16 to 21 with Part 3, and 22 to 25 ask you to explain.

Part 1: measuring and plotting

A ladybug is \displaystyle \frac{3}{8} inch long. Is it longer or shorter than \displaystyle \frac{1}{4} inch, and by how much?
Write each length in eighths of an inch: \displaystyle \frac{1}{2}, \displaystyle 1\frac{1}{4}, \displaystyle 2\frac{3}{4}.
A seashell measures \displaystyle 1\frac{7}{8} inches to the nearest eighth. What is its length to the nearest half inch?
On an earlier walk, Kagiso measured 9 pinecones to the nearest half inch: 3, \displaystyle 2\frac{1}{2}, \displaystyle 3\frac{1}{2}, 3, 4, 3, \displaystyle 2\frac{1}{2}, \displaystyle 3\frac{1}{2}, 3 inches. How many dots go above each mark of his line plot?
Akua’s friend wrote down 7 shell lengths: \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{3}{8}, \displaystyle 1\frac{2}{8}, \displaystyle 1\frac{1}{2}, \displaystyle 1\frac{4}{8}, \displaystyle 1\frac{5}{8}, \displaystyle 1\frac{3}{8} inches. On a line plot marked in eighths, how many dots go above each mark?
The shortest of a set of lengths is \displaystyle \frac{3}{8} inch and the longest is \displaystyle 1\frac{1}{8} inches. A line plot runs from \displaystyle \frac{1}{4} to \displaystyle 1\frac{1}{4}, marked every eighth of an inch. How many marks does it have, counting both ends?
Laleh’s line plot is marked every quarter inch from 0 to 1. Then a ladybug measures \displaystyle \frac{3}{8} inch. Where should its dot go, and what should Laleh do?

Part 2: reading the plot

Exercises 8 to 12 and 15 use Akua’s seashell plot from Example 1.

What is the most common seashell length, and how many shells have it?
What is the difference between the longest and the shortest shell?
What is the total length of the shells at \displaystyle 1\frac{3}{4} inches?
How many shells are longer than \displaystyle 1\frac{1}{2} inches? How many are shorter than 2 inches?
Akua lines up all 10 shells end to end. How long is the row?
In the ladybug plot from Example 2, how much longer is the total length of the \displaystyle \frac{3}{8} inch ladybugs than the total length of the \displaystyle \frac{1}{4} inch ladybugs?
In the pinecone plot from Example 3, what is the total length of the pinecones that are \displaystyle 2\frac{1}{2} inches or shorter? Using Example 4, what is the total length of all 11 pinecones?
Akua puts her two shortest shells end to end. Is that longer or shorter than her longest shell, and by how much?

Part 3: sharing the total equally

If Akua’s 10 seashells from Example 1 were all the same length, with the same total as in Exercise 12, how long would each shell be?
Four beakers hold \displaystyle \frac{1}{8}, \displaystyle \frac{3}{8}, \displaystyle \frac{5}{8} and \displaystyle \frac{7}{8} liter of water. If the water were shared equally, how much would each beaker hold?
For the beakers in Exercise 17, which beakers give water away and which take it in, and how much? Show that the amount given equals the amount taken.
Kagiso’s 9 pinecones from Exercise 4 have the same total length as 9 pinecones that are all the same length. How long is each of those?
Akua has 5 shells. If their total were shared equally, each would be \displaystyle 1\frac{1}{2} inches. Four of them are \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2}, \displaystyle 1\frac{3}{4} and \displaystyle 1\frac{1}{4} inches. How long is the fifth?
The class measures 4 more ladybugs, each \displaystyle \frac{1}{2} inch, and adds them to the 12 from Example 2. What is the equal share of the total length now?

Reasoning

Kagiso labels the marks of a line plot 2, \displaystyle 2\frac{1}{4}, \displaystyle 2\frac{1}{2}, 3, \displaystyle 3\frac{1}{4}, spaced evenly. Explain what is wrong and how to fix it.
Explain why a length written as \displaystyle \frac{2}{8} inch and a length written as \displaystyle \frac{1}{4} inch get their dots above the same mark.
Explain why the equal share of a set of measurements can never be longer than the longest one or shorter than the shortest one.
Akua says, “The most common length and the equal share are always the same number.” Use one of the plots in this lesson to show she is wrong.

Answers

Longer, by \displaystyle \frac{1}{8} inch. \displaystyle \frac{1}{4} = \frac{2}{8}, and \displaystyle \frac{3}{8} - \frac{2}{8} = \frac{1}{8}.
\displaystyle \frac{1}{2} = \frac{4}{8}, \displaystyle 1\frac{1}{4} = 1\frac{2}{8}, \displaystyle 2\frac{3}{4} = 2\frac{6}{8}. Multiply the top and bottom of each fraction part by the same number (4 for halves, 2 for quarters).
2 inches. The half inch marks on each side are \displaystyle 1\frac{1}{2} = 1\frac{4}{8} and 2. The shell is \displaystyle \frac{3}{8} inch past \displaystyle 1\frac{1}{2} but only \displaystyle \frac{1}{8} inch short of 2, so 2 is nearer.
2 dots above \displaystyle 2\frac{1}{2}, 4 above 3, 2 above \displaystyle 3\frac{1}{2} and 1 above 4. Check: 2 + 4 + 2 + 1 = 9.
2 dots above \displaystyle 1\frac{2}{8}, 2 above \displaystyle 1\frac{3}{8}, 2 above \displaystyle 1\frac{4}{8} and 1 above \displaystyle 1\frac{5}{8}. \displaystyle 1\frac{1}{4} = 1\frac{2}{8} and \displaystyle 1\frac{1}{2} = 1\frac{4}{8}, so each shares a mark with its other name. Check: 2 + 2 + 2 + 1 = 7.
9 marks. In eighths the line runs from \displaystyle \frac{2}{8} to \displaystyle \frac{10}{8}: the marks are \displaystyle \frac{2}{8}, \frac{3}{8}, \ldots, \frac{10}{8}, which is 10 - 2 + 1 = 9 marks.
There is no mark for \displaystyle \frac{3}{8}: it falls halfway between the \displaystyle \frac{1}{4} and \displaystyle \frac{1}{2} marks. Laleh should mark the line every eighth of an inch instead. The dots already there stay where they are, because every quarter mark is also an eighth mark (\displaystyle \frac{1}{4} = \frac{2}{8}, \displaystyle \frac{1}{2} = \frac{4}{8}, \displaystyle \frac{3}{4} = \frac{6}{8}), and the new dot goes above \displaystyle \frac{3}{8}.
\displaystyle 1\frac{1}{2} inches, with 4 shells. That mark has the tallest stack.
1 inch. \displaystyle 2\frac{1}{4} - 1\frac{1}{4} = 1.
\displaystyle 3\frac{1}{2} inches. \displaystyle 2 \times 1\frac{3}{4} = 2\frac{6}{4} = 3\frac{2}{4} = 3\frac{1}{2}.
4 shells are longer than \displaystyle 1\frac{1}{2} inches (2 + 1 + 1). 8 shells are shorter than 2 inches (2 + 4 + 2).
\displaystyle 16\frac{1}{4} inches. The totals at the marks are \displaystyle 2 \times 1\frac{1}{4} = 2\frac{1}{2}, \displaystyle 4 \times 1\frac{1}{2} = 6, \displaystyle 2 \times 1\frac{3}{4} = 3\frac{1}{2}, then 2 and \displaystyle 2\frac{1}{4}. Adding, \displaystyle 2\frac{1}{2} + 3\frac{1}{2} = 6, and \displaystyle 6 + 6 + 2 + 2\frac{1}{4} = 16\frac{1}{4}.
\displaystyle \frac{7}{8} inch longer. \displaystyle 5 \times \frac{3}{8} = \frac{15}{8} and \displaystyle 4 \times \frac{2}{8} = \frac{8}{8}, and \displaystyle \frac{15}{8} - \frac{8}{8} = \frac{7}{8}.
\displaystyle 14\frac{3}{8} inches for the 6 pinecones of \displaystyle 2\frac{1}{2} inches or shorter. The totals at the marks are \displaystyle 2\frac{1}{8}, then \displaystyle 2 \times 2\frac{3}{8} = 4\frac{6}{8}, then \displaystyle 3 \times 2\frac{1}{2} = 7\frac{4}{8}, and \displaystyle 2\frac{1}{8} + 4\frac{6}{8} + 7\frac{4}{8} = 13\frac{11}{8}, which is \displaystyle 14\frac{3}{8}. All 11 pinecones: \displaystyle 14\frac{3}{8} + 14 = 28\frac{3}{8} inches.
Longer, by \displaystyle \frac{1}{4} inch. \displaystyle 1\frac{1}{4} + 1\frac{1}{4} = 2\frac{2}{4} = 2\frac{1}{2}, and \displaystyle 2\frac{1}{2} - 2\frac{1}{4} = \frac{1}{4}.
\displaystyle 1\frac{5}{8} inches. \displaystyle 16\frac{1}{4} = 16\frac{2}{8} = \frac{130}{8}, and 130 eighths shared among 10 shells is 13 eighths each: \displaystyle \frac{13}{8} = 1\frac{5}{8}.
\displaystyle \frac{1}{2} liter. The total is \displaystyle \frac{1 + 3 + 5 + 7}{8} = \frac{16}{8} = 2 liters, and \displaystyle 2 \div 4 = \frac{2}{4} = \frac{1}{2}.
The equal share is \displaystyle \frac{1}{2} = \frac{4}{8}. The \displaystyle \frac{7}{8} beaker gives \displaystyle \frac{3}{8} and the \displaystyle \frac{5}{8} beaker gives \displaystyle \frac{1}{8}, so \displaystyle \frac{4}{8} is given. The \displaystyle \frac{1}{8} beaker takes \displaystyle \frac{3}{8} and the \displaystyle \frac{3}{8} beaker takes \displaystyle \frac{1}{8}, so \displaystyle \frac{4}{8} is taken. Given equals taken.
\displaystyle 3\frac{1}{9} inches. The total is \displaystyle 2 \times 2\frac{1}{2} + 4 \times 3 + 2 \times 3\frac{1}{2} + 4 = 5 + 12 + 7 + 4 = 28 inches, and \displaystyle 28 \div 9 = \frac{28}{9} = 3\frac{1}{9}.
\displaystyle 1\frac{3}{4} inches. The total of all 5 is \displaystyle 5 \times 1\frac{1}{2} = 7\frac{1}{2}. The four known shells total \displaystyle 1\frac{1}{4} + 1\frac{1}{2} + 1\frac{3}{4} + 1\frac{1}{4} = 5\frac{3}{4}, and \displaystyle 7\frac{1}{2} - 5\frac{3}{4} = 1\frac{3}{4}.
\displaystyle \frac{3}{8} inch. The total is \displaystyle 4 + 4 \times \frac{1}{2} = 6 inches for 16 ladybugs, and \displaystyle 6 \div 16 = \frac{6}{16} = \frac{3}{8}.
The mark \displaystyle 2\frac{3}{4} is missing. The step from \displaystyle 2\frac{1}{2} to 3 is \displaystyle \frac{1}{2} inch, but it is drawn the same size as the \displaystyle \frac{1}{4} inch steps, so equal distances no longer mean equal amounts. Fix it by adding a mark for \displaystyle 2\frac{3}{4}, so every step is \displaystyle \frac{1}{4} inch.
\displaystyle \frac{2}{8} and \displaystyle \frac{1}{4} are the same number: cutting each quarter into 2 equal pieces makes eighths, and one quarter is then 2 eighths. A number has exactly one point on the number line, so both lengths are the same point, and both dots go above that mark.
If every measurement were made the same as the longest one, the total would grow or stay the same, never shrink. So the total is at most the longest times the number of measurements, and the equal share, the total divided by that number, is at most the longest. In the same way, making every value the same as the shortest can only shrink the total, so the equal share is at least the shortest.
The ladybug plot of Example 2. Its most common length is \displaystyle \frac{3}{8} inch (the stack of 5), but its equal share is \displaystyle \frac{1}{3} inch (Example 7), and \displaystyle \frac{1}{3} = \frac{8}{24} is less than \displaystyle \frac{3}{8} = \frac{9}{24}. The beaker plot of Example 6 is another: its most common amount is \displaystyle \frac{1}{4} liter, its equal share \displaystyle \frac{3}{8} liter.