Grade 2 · 2.MD.D.9 · 4.MD.B.4

Reading dot plots: clusters, gaps and peaks

Twelve worms from a garden dig have been measured, and their lengths sit in a list of twelve numbers. What picture lets you see at a glance how long the worms are, which length came up most, and how far apart the shortest and longest are?

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Making a dot plotGrade 2 Finding the medianMath 6

Where you are 3 in Dot plots

Line plots with fractions

When you reach for it

Tasi digs up a garden bed and finds 12 worms. She lets each one stretch out along a ruler and writes down its length to the nearest inch: 5, 3, 6, 5, 4, 9, 6, 5, 7, 4, 5, 6. Every worm is on that list, but the list does not answer much. Which length came up most? How long was the longest worm? Are most of the worms about the same size? To answer, you would have to sort the list and count it again, one number at a time.

A line plot shows all twelve lengths at once. It is a number line, like the one on a ruler, with one dot above it for each worm, placed above that worm’s length. When two worms have the same length, their dots stack up. The picture shows the whole list, already sorted and already counted. The same display is also called a dot plot. The two names mean the same thing, and this lesson says line plot.

A line plot is a close cousin of the picture graph and the bar graph, but it is used for a different kind of list. A picture graph counts things in groups that have names, such as favorite fruits. A line plot counts measurements, which are numbers, so the groups sit in number order along a number line, and the space between them means something. That is what lets a line plot answer questions like “how many worms are longer than 5 inches?” and “how far apart are the shortest and the longest?”

Reach for a line plot whenever many things of one kind have been measured with the same unit and you want to see the measurements together. This lesson reads line plots; drawing one from a list is Making a dot plot. It takes three days. Part 1 reads the stacks and finds how many in all. Part 2 asks the plot questions: longer, shorter, most common, longest and shortest. Part 3 looks at the shape: where the dots bunch up, where there is an empty stretch, and where the tallest stack stands.

The procedure

Part 1: one dot for each thing measured

Figure 1Tasi’s 12 worms, one dot for each worm, stacked above its length.

The number line. The line along the bottom is a number line marked at every whole number. The marks are spaced equally, just like the inch marks on a ruler, because each step from one mark to the next is one more inch (Length on a number line). The words under the line, “worm length (inches)”, say what was measured and in which unit. Without them, the 5 could mean 5 inches, 5 centimeters or 5 of anything.

One dot for each worm. Each worm was measured once (Measuring length) and got exactly one dot, placed above the mark for its length. A worm 5 inches long has its dot above the 5. When a second worm is also 5 inches long, its dot goes on top of the first, and so on. So the dots above a mark form a stack, and the stack is all the worms with that length.

Reading a stack. Count the dots above a mark and you know how many things had that length. There are 4 dots above the 5, so 4 worms are 5 inches long. A mark with no dots above it is not a mistake. It says that nothing had that length: no worm is 8 inches long. Its count is 0.

How many in all. Add up the stacks:

1 + 2 + 4 + 3 + 1 + 0 + 1 = 12

So the plot holds 12 worms. This works because every worm has exactly one dot, and every dot belongs to exactly one worm. No worm was skipped and no worm was drawn twice, so counting the dots is the same as counting the worms. It is also a check: Tasi’s list has 12 numbers, so her plot must have 12 dots.

Back to the list. The plot can be read back into a list. Go along the line from left to right and write each length as many times as its stack is tall: one 3, two 4s, four 5s, and so on. That gives the same twelve numbers as Tasi’s list, sorted from shortest to longest. What the plot cannot say is which worm is which, or the order in which they were dug up. It keeps the lengths and forgets everything else.

Dots or Xs. Some line plots use an X for each thing instead of a dot. It is read the same way: one X, one thing measured.

Why the empty marks stay. The marks at 2, 8 and 10 have no dots, but they stay on the line. Leaving out the 8 would push the 9 next to the 7 and make a 9-inch worm look only one step longer than a 7-inch worm, when it is two inches longer. Keeping every mark keeps the line honest, the same way a ruler never skips an inch.

Part 2: asking the plot questions

The number line puts the lengths in order. Shorter lengths are to the left and longer ones to the right, just as on a ruler. Every question in this part comes from that.

Longer than, shorter than. To count the worms longer than 5 inches, count every dot to the right of the 5 mark. The dots above the 5 itself are not longer than 5 inches, so they are left out. To count the worms shorter than 5 inches, count every dot to the left of the 5. A question like “5 inches or longer” includes the stack above the 5 as well. The words matter, because the stack on the mark itself belongs to one count and not the other.

The most common length. The tallest stack has the most dots, so its length came up most often. In the worm plot the tallest stack is above the 5, with 4 dots, so the most common length is 5 inches. Its height is the number of worms, 4.

How many more. To compare two lengths, compare their stacks. There are 4 worms at 5 inches and 3 at 6 inches, and 4 - 3 = 1, so one more worm is 5 inches long than 6 inches long.

Longest and shortest. The dot farthest to the right is the longest thing measured, and the dot farthest to the left is the shortest. Empty marks at the ends do not count: the line runs from 2 to 10, but the longest worm is 9 inches, because there is no dot at 10.

How far apart. The longest worm and the shortest are as far apart as their marks on the number line. Count the jumps from one mark to the other, or subtract the smaller length from the larger. Both give the same answer, since each jump on the line is one inch (Length on a number line).

Part 3: clusters, gaps and peaks

Step back from single stacks and look at the whole picture. Three plain words describe what you see.

A cluster is a place where the dots bunch up: stacks standing side by side, with no empty mark between them. In the worm plot the stacks from 3 to 7 all have dots, so the worms from 3 to 7 inches form a cluster. A cluster says “most of the things have about this size”.

A gap is an empty stretch: one or more marks with no dots, with dots on both sides. The worm plot has a gap at 8 inches. Empty marks at the very ends of the line, like the 2 and the 10, are not called gaps, because there is nothing on the other side of them; they just show where the line was drawn.

A peak is the tallest stack, the top of the hill the dots make. The worm plot’s peak is at 5 inches. The peak is the most common length from Part 2, seen as a shape. A plot can have two hills, each with its own top, and then it is fair to say it has two peaks.

Figure 2The worms bunch up from 3 to 7 inches, the tallest stack is at 5, and nothing is at 8.

The worm with the dot at 9 sits by itself, past the gap. A dot that stands apart from the rest like this is worth a second look. Maybe it was a very big worm; maybe someone misread the ruler. The plot cannot say which, but it shows where to look.

These words are descriptions, not exact rules. Two people might draw a cluster a little wider or a little narrower, and that is fine as long as each can point to the dots they mean. What matters is saying, in words, what the picture shows: where most of the things are, where none are, and which length is the most common.

Worked examples

Example 1

Part 1: how many worms of each length

Use Tasi’s worm plot (Figure 1). How many worms are 6 inches long? How many are 8 inches long? How many worms did Tasi measure in all?

Solution

Each worm has one dot above its length, so the stack above a mark is the worms with that length. There are 3 dots above the 6, so 3 worms are 6 inches long. There are no dots above the 8, so 0 worms are 8 inches long: no worm was that length.

To find how many worms in all, add the stacks from left to right, including the empty one at 8:

1 + 2 + 4 + 3 + 1 + 0 + 1 = 12

Tasi measured 12 worms. Her list also has 12 numbers, which checks the plot: every worm has a dot and no worm has two.

Example 2

Part 1: reading Xs back into a list

Zyanya measures the crayons in her box to the nearest centimeter and makes the line plot in Figure 3, with an X for each crayon. How many crayons are in the box? Write every crayon’s length, from shortest to longest.

Figure 3Zyanya’s crayons, drawn with an X for each crayon instead of a dot.

Solution

An X is read just like a dot: one X, one crayon. The stacks are 2 at 3 centimeters, 3 at 4, 2 at 5, 1 at 8 and 4 at 9. The marks at 2, 6, 7 and 10 are empty. Adding the stacks,

2 + 3 + 2 + 1 + 4 = 12

so there are 12 crayons.

To write the list, go along the line from left to right and write each length once for every X above it: 3, 3, 4, 4, 4, 5, 5, 8, 9, 9, 9, 9. The list has 12 numbers, one for each X, and it comes out already sorted, because the number line is in order. The plot does not say which crayon is red or blue; it only keeps the lengths.

Example 3

Part 2: longer than and shorter than

Use Tasi’s worm plot (Figure 1). How many worms are longer than 5 inches? How many are 5 inches or shorter?

Solution

Longer lengths are to the right on the number line. The worms longer than 5 inches are the dots to the right of the 5 mark, at 6, 7, 8 and 9. The stack above the 5 is not counted, because a 5-inch worm is not longer than 5 inches.

3 + 1 + 0 + 1 = 5

So 5 worms are longer than 5 inches.

The worms 5 inches or shorter are the dots at 5 and to its left, at 3, 4 and 5:

1 + 2 + 4 = 7

So 7 worms are 5 inches or shorter. The two groups do not share a worm, and together they hold every worm, so they should make the whole: 5 + 7 = 12, which matches Example 1.

Example 4

Part 2: most common, longest and shortest

Use Tasi’s worm plot again. What is the most common worm length? How many more worms are 5 inches long than 4 inches long? How long are the longest and the shortest worms, and how far apart are those two lengths?

Solution

The tallest stack is above the 5, with 4 dots, so the most common length is 5 inches.

There are 4 worms at 5 inches and 2 worms at 4 inches, so

4 - 2 = 2

and 2 more worms are 5 inches long than 4 inches long.

The dot farthest to the right is above the 9, so the longest worm is 9 inches long. The mark at 10 is empty and does not count. The dot farthest to the left is above the 3, so the shortest worm is 3 inches long. To see how far apart they are, count the jumps along the line from 3 to 9: 4, 5, 6, 7, 8, 9, which is 6 jumps of one inch each. Subtracting says the same thing:

9 - 3 = 6

The longest worm is 6 inches longer than the shortest.

Example 5

Part 3: the shape of the worm plot

Describe the worm plot in words. Where do the dots bunch up? Is there a gap? Where is the peak?

Solution

Look at Figure 2. The stacks at 3, 4, 5, 6 and 7 stand side by side with no empty mark between them, so they form a cluster from 3 to 7 inches. It holds

1 + 2 + 4 + 3 + 1 = 11

of the 12 worms. The mark at 8 is empty with dots on both sides, so there is a gap at 8 inches. Past the gap, one worm sits alone at 9 inches. The tallest stack, 4 dots, is at 5 inches, so that is the peak.

In plain words: most of Tasi’s worms were between 3 and 7 inches long, the most common length was 5 inches, and one worm, at 9 inches, was longer than all the rest by at least two inches.

Example 6

Part 3: two clusters

Here is Zyanya’s crayon plot again, drawn with dots and with its shape marked. Describe the clusters, the gap and the peaks. What might they say about the crayons?

Figure 4Two clusters with an empty stretch between them. Each cluster has its own peak.

Solution

The dots bunch up in two places. The stacks at 3, 4 and 5 centimeters form one cluster, with 2 + 3 + 2 = 7 crayons. The stacks at 8 and 9 form another, with 1 + 4 = 5 crayons. Between them, the marks 6 and 7 are both empty, so there is a gap from 6 to 7 centimeters.

Each cluster has its own tallest stack. In the short cluster it is at 4, with 3 crayons; in the long cluster it is at 9, with 4 crayons. So the plot has two peaks. The taller one is at 9, which makes 9 centimeters the most common length.

The shape tells a story the list hides. A crayon gets shorter each time it is used and sharpened. Two clusters with an empty stretch between them suggest two kinds of crayon in the box: some nearly new, about 8 or 9 centimeters long, and some worn down to about 3 to 5 centimeters, with none in between. The plot cannot prove that story, but it is what the shape suggests, and Zyanya can check by looking in the box.

Notice also that the most common length, 9 centimeters, is not where most of the crayons are: only 4 of the 12 are 9 centimeters long, while the short cluster holds 7.

Example 7

Part 3: one more worm

Tasi digs a little deeper and finds one more worm. It is 2 inches long. She adds its dot to her plot. How many worms are on the plot now? What are the longest and shortest lengths, and how far apart are they? Does the peak move? What happens to the cluster?

Solution

The new worm gets one new dot, above the 2. Every other dot stays where it was.

There were 12 worms and there is 1 more, so

12 + 1 = 13

and the plot now has 13 dots.

The new dot is farther left than any other, so the shortest worm is now 2 inches long. Nothing new was added on the right, so the longest is still 9 inches. The two lengths are now

9 - 2 = 7

inches apart, one more than before, because the left end moved one mark farther out.

The new dot starts a stack of 1 at 2, and the stack at 5 still has 4 dots, more than any other, so the peak stays at 5 inches. The mark at 2 sits right next to the 3, with no empty mark between, so the new worm joins the cluster: it now runs from 2 to 7 inches and holds 12 of the 13 worms. The gap at 8 and the lone worm at 9 have not changed.

One new dot can change the shortest and the longest, because those depend on only the end dots. It takes much more to move the peak, which depends on the heights of the stacks.

Boundaries

A line plot is for numbers. The marks along the bottom are a number line, so the things measured must have number values, like lengths in inches. Favorite fruits or colors of crayons have no number order and no distance between them, so they go in a picture graph or a bar graph instead.
One unit for everything. Every measurement on one plot uses the same unit. A 5-centimeter crayon and a 5-inch crayon cannot share a mark, because they are not the same length.
Each dot is rounded to its mark. Lengths here are to the nearest whole inch or centimeter, so two worms whose dots share a stack may differ by a little. A plot can be no more exact than the measuring was. Marks at halves and quarters of an inch come later, in Line plots with fractions.
The plot keeps how many, not which. It shows how many things had each length, but not which thing had which length, nor the order they were measured in.
The most common length need not be one length. If two stacks tie for tallest, both lengths are the most common. If every stack has the same height, no length is more common than another.
Cluster, gap and peak are describing words. They have no exact rule. A gap needs dots on both sides; past that, where a cluster begins and ends is a judgment about the picture, and the answer should point to the dots it means.

Where this goes next

The very next step is drawing a line plot from a list of measurements, Making a dot plot, which runs this lesson backward: each length in the list becomes one dot above its mark. A line plot also sits beside the bar graph: a stack of dots is a bar made of separate pieces, and both show how many by how tall.

Later the marks on the line get finer. In Line plots with fractions the number line is marked in halves, quarters and eighths of an inch, and the plot is used to add and share out the measurements. In middle school, line plots of larger data sets are where the median is read off by counting dots in from both ends, and the plain words of Part 3 become the language of shape: one hill, a flat spread, or the two-peaked shape of Zyanya’s crayons, which is studied in Uniform and bimodal shapes. The habit that starts here, looking at where the dots bunch up before doing any arithmetic, carries all the way through.

Exercises

Part 1: reading the stacks

Use Ebun’s maple seed plot for Exercises 1 to 4.

Figure 5Ebun’s maple seeds, for Exercises 1 to 4, 6 to 11, 15 to 17 and 22.

How many maple seeds are 3 centimeters long?
How many maple seeds are 6 centimeters long?
How many maple seeds did Ebun measure in all?
Write the length of every seed, from shortest to longest.
Tasi measures seven worms from a second garden bed. Their lengths in inches are 4, 6, 5, 4, 7, 5, 4. On a line plot of these worms, how many dots would sit above the 4? How many above the 6? How many dots would the plot have in all?

Part 2: asking the plot questions

Use Ebun’s maple seed plot for Exercises 6 to 11.

How many seeds are longer than 4 centimeters?
How many seeds are shorter than 4 centimeters?
How many seeds are 4 centimeters or longer?
What is the most common seed length? How many seeds have that length?
How long are the longest seed and the shortest seed? How far apart are those two lengths?
How many more seeds are 4 centimeters long than 5 centimeters long?
Use Zyanya’s crayon plot (Figure 3). How many crayons are shorter than 9 centimeters?
Use Tasi’s worm plot (Figure 1). How many worms are 4 or 5 inches long?
Use Tasi’s worm plot. How many worms are longer than 3 inches and shorter than 7 inches?

Part 3: clusters, gaps and peaks

Use Ebun’s maple seed plot for Exercises 15 to 17.

Where is the peak of the seed plot?
Where is there a gap in the seed plot?
Where do the dots bunch up? How many seeds are in that cluster?

Use Ebun’s worm plot for Exercises 18 to 21.

Figure 6Ebun’s worms from a second dig, for Exercises 18 to 21.

How many worms did Ebun measure?
Describe the clusters and the gap in Ebun’s worm plot.
What is the most common worm length in Ebun’s plot?
How far apart are the longest and the shortest of Ebun’s worms?

Explain

Are most of Ebun’s maple seeds 4 centimeters long? How do you know?
Tasi’s worm plot has a mark at 8 with no dots above it. Why keep that mark on the line?
A line plot has 10 dots. Can its tallest stack have only 1 dot? What would the plot look like?
Why must the number of dots on a line plot be the same as the number of things measured?

Answers

4 seeds. There are 4 dots above the 3.
0 seeds. The mark at 6 has no dots above it.
15 seeds. Adding the stacks: 1 + 4 + 6 + 3 + 1 = 15.
2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 7. Each length is written once for every dot above it, going from left to right. The list has 15 numbers, one for each dot.
3 dots above the 4, 1 dot above the 6, and 7 dots in all. The 4 appears three times in the list and the 6 once; each worm gets one dot, so the plot has as many dots as the list has numbers.
4 seeds. The dots to the right of the 4 mark are at 5 and 7: 3 + 1 = 4.
5 seeds. The dots to the left of the 4 mark are at 2 and 3: 1 + 4 = 5.
10 seeds. This time the stack at 4 counts too: 6 + 3 + 1 = 10. Check: 5 + 10 = 15, every seed.
4 centimeters, with 6 seeds. The tallest stack is above the 4.
The longest is 7 centimeters and the shortest is 2 centimeters. They are 7 - 2 = 5 centimeters apart, five jumps along the line.
3 more. There are 6 seeds at 4 centimeters and 3 at 5: 6 - 3 = 3.
8 crayons. The Xs to the left of the 9 mark are at 3, 4, 5 and 8: 2 + 3 + 2 + 1 = 8. Check: 8 shorter and 4 at 9 make 12.
6 worms. The stacks at 4 and 5: 2 + 4 = 6.
9 worms. The lengths between 3 and 7, not counting 3 or 7 themselves, are 4, 5 and 6: 2 + 4 + 3 = 9.
At 4 centimeters. That stack has 6 dots, more than any other.
At 6 centimeters. The mark at 6 is empty, and there are dots on both sides of it, at 5 and at 7. The empty marks at 1 and 8 are at the ends of the line, so they are not gaps.
From 2 to 5 centimeters, with 1 + 4 + 6 + 3 = 14 seeds. The stacks at 2, 3, 4 and 5 stand side by side, and only the one seed at 7 is outside the cluster, past the gap.
12 worms: 2 + 3 + 1 + 2 + 3 + 1 = 12.
Two clusters and one gap. One cluster runs from 2 to 4 inches, with 2 + 3 + 1 = 6 worms. The other runs from 8 to 10 inches, with 2 + 3 + 1 = 6 worms. Between them the marks 5, 6 and 7 are all empty, so there is a gap from 5 to 7 inches. (The empty marks at 1 and 11 are ends, not gaps.)
Both 3 inches and 9 inches. Those two stacks tie for tallest, with 3 worms each, so the plot has two peaks and two most common lengths.
8 inches. The shortest worm is 2 inches and the longest is 10 inches, and 10 - 2 = 8.
No. Only 6 of the 15 seeds are 4 centimeters long. The other 15 - 6 = 9 seeds have other lengths, and 9 is more than 6. The 4 is the most common length, but most of the seeds are not that length.
The marks must stay equally spaced, like the inch marks on a ruler, so that the distance between two dots shows how far apart their lengths are. Without the 8, the 9 would sit next to the 7 and look only one inch longer. The empty mark also tells you something: no worm was 8 inches long.
Yes. If all 10 things have different lengths, each stack has 1 dot, so the tallest stack has 1 dot. The plot is a flat row of single dots above 10 different marks. It has no single peak, and no length is more common than another.
Each thing measured gets exactly one dot, above its own length, and every dot was put there for one thing. No thing is left out and none is drawn twice, so counting the dots counts the things.