Line Plots: Share Equally, Solve, and Check | Grade 5 Math | LevelKite

You already know how to make a line plot and how to read one. Each mark stands for one measurement. The marks sit above the scale, which is the number line at the bottom.

In this lesson you put all the marks to work together. You will share a total equally, solve problems that take more than one step, and check line plot work for mistakes.

Here is the data you will use first. A class collected rainwater in 8 jars that are all the same size. The line plot below shows how many liters of water are in each jar.

Line plot

Each mark is one jar. The 3 marks above

\frac{1}{2}

are 3 jars that hold

\frac{1}{2}

liter each.

Read the scale first. It is split into equal spaces, and each space is $\frac{1}{4}$ liter. Count spaces from $0$ to name any value: 2 spaces is $\frac{2}{4}$ , or $\frac{1}{2}$ liter, and 6 spaces is $\frac{6}{4}$ , or $1\frac{1}{2}$ liters.

Now look at the stack above $\frac{1}{2}$ . A stack is the pile of marks above one value. This stack has 3 marks, so 3 jars hold $\frac{1}{2}$ liter each. You can show the water in that stack three ways:

On the plot: 3 marks above $\frac{1}{2}$
In words: 3 jars with $\frac{1}{2}$ liter in each
In numbers: $\frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 3 \times \frac{1}{2} = \frac{3}{2} = 1\frac{1}{2}$ liters

No marks sit above $\frac{3}{4}$ or $1\frac{1}{4}$ , so no jar holds those amounts.

Key concept

Every mark is one measurement

Each mark on a line plot is one measurement. A stack of marks is the same measurement repeated, once for every mark in the stack. So whenever you add up the data, share it equally, or check someone's work, count every mark, not just every stack.

\text{amount in a stack} = \text{number of marks} \times \text{value under the stack}

What does it mean to share the water equally? Picture pouring all 8 jars into one big tub. Then you pour the water back so that every jar gets the same amount. That amount is the equal share. Sharing out again like this is called redistributing the water.

No water is added and none is spilled, so the total stays the same. Only the amount in each jar changes. That gives you a plan: find the total of all the measurements, then divide it by the number of measurements.

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

Worked example

Share the water equally

The line plot shows 8 jars of water. Two jars hold $\frac{1}{4}$ liter, three hold $\frac{1}{2}$ liter, one holds $1$ liter and two hold $1\frac{1}{2}$ liters. If all the water is redistributed equally, how much water will each jar hold?

Start with the two stacks on the left: 2 marks above one fourth and 3 marks above one half. A stack is one amount repeated once for each mark, so multiply the amount by the number of marks. Rename each answer so it is easy to add later.
$2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \qquad 3 \times \frac{1}{2} = \frac{3}{2} = 1\frac{1}{2}$
Do the same for the two stacks on the right. The stack above 1 has 1 mark. The stack above one and one half has 2 marks, and two of that amount make 3 whole liters.
$1 \times 1 = 1 \qquad 2 \times 1\frac{1}{2} = 1\frac{1}{2} + 1\frac{1}{2} = 3$
Add the four stack amounts. Every jar is in exactly one stack, so this sum is all the water. Add the two amounts with halves first, because together they make 2 whole liters.
$\frac{1}{2} + 1\frac{1}{2} + 1 + 3 = 2 + 1 + 3 = 6$
Count the marks to find the number of jars, because each mark is one jar. Then divide the total by that number. Sharing equally among 8 jars splits the 6 liters into 8 equal parts. Each part is six eighths of a liter, which is the same as three fourths of a liter.
$2 + 3 + 1 + 2 = 8 \qquad 6 \div 8 = \frac{6}{8} = \frac{3}{4}$
Check with multiplication. Redistributing does not change the total, so 8 jars with three fourths of a liter in each must still make 6 liters.
$8 \times \frac{3}{4} = \frac{24}{4} = 6$

Answer

Each jar will hold $\frac{3}{4}$ liter of water. A new line plot of the jars would show all 8 marks in one stack above $\frac{3}{4}$ , a value that had no marks before.

Worked example

Compare two groups of jars

Use the same line plot of the 8 jars. The class pours every jar that holds less than $1$ liter into a blue bucket. They pour the two fullest jars into a red bucket. Which bucket gets more water, and how much more?

Choose the stacks for the blue bucket. Less than 1 liter means to the left of 1 on the scale. That is the stack of 2 marks above one fourth and the stack of 3 marks above one half. The jar with exactly 1 liter is left out, because 1 is not less than 1. Multiply each amount by its number of marks.
$2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \qquad 3 \times \frac{1}{2} = \frac{3}{2} = 1\frac{1}{2}$
Both stacks go into the blue bucket, so add their amounts.
$\frac{1}{2} + 1\frac{1}{2} = 2$
Now find the water in the red bucket. The two fullest jars are the 2 marks farthest to the right, above one and one half. Multiply that amount by 2.
$2 \times 1\frac{1}{2} = 3$
Compare the buckets. 3 liters is more than 2 liters, so the red bucket has more. Subtract to find how much more.
$3 - 2 = 1$

Answer

The red bucket gets more water. It holds $3$ liters and the blue bucket holds $2$ liters, so the red bucket holds $1$ liter more. It has more water even though it was filled from fewer jars.

Now you be the checker. When you look at line plot work, ask three questions:

Is every measurement on the plot? Count the marks. The count must match the number of measurements in the list.
Is the scale read correctly? Find the size of one space on the scale before you name any value.
Is every mark used? A stack counts once for each of its marks, as you just saw.

Try the questions on this work. Kai measured 8 seeds to the nearest $\frac{1}{8}$ inch. Here is his list of lengths, in inches:

$\frac{3}{8}$ , $\frac{5}{8}$ , $\frac{3}{8}$ , $\frac{7}{8}$ , $\frac{5}{8}$ , $\frac{3}{8}$ , $\frac{7}{8}$ , $\frac{5}{8}$

Kai made the line plot below from his list.

Line plot

Kai's line plot: 2 marks above

\frac{3}{8}

, 3 marks above

\frac{5}{8}

and 2 marks above

\frac{7}{8}

. Check it against his list of 8 lengths.

Worked example

Find and fix the mistakes

Kai's list of 8 seed lengths and his line plot are shown above. His partner Mia used the plot to find the total length of the shortest seeds. Mia wrote: "The shortest seeds are 3 spaces from 0, so each is $\frac{3}{4}$ inch long. There are 2 marks, so the total is $2 \times \frac{3}{4} = 1\frac{1}{2}$ inches." Find the mistakes in the plot and in Mia's work, and correct them.

Check that every measurement is on the plot. The list has 8 lengths, so the plot needs 8 marks. Count the marks in the three stacks. The plot is one mark short, so one measurement was left off.
$2 + 3 + 2 = 7$
Find the missing measurement by matching each value in the list with its stack. Five eighths is in the list 3 times and has 3 marks. Seven eighths is in the list 2 times and has 2 marks. Three eighths is in the list 3 times but has only 2 marks. Fix the plot by adding 1 mark above three eighths. Now the plot has as many marks as the list has lengths.
$7 + 1 = 8$
Check how Mia read the scale. She counted each space as one fourth of an inch. Count the spaces from 0 to 1 to test that. There are 8 equal spaces, so each space is one eighth of an inch. The shortest seeds are 3 spaces from 0, so each is three eighths of an inch long.
$3 \times \frac{1}{8} = \frac{3}{8}$
Redo the total with both fixes. Mia was right to multiply by the number of marks, because a stack is one measurement repeated. Use the corrected stack of 3 marks and the correct length of three eighths of an inch.
$3 \times \frac{3}{8} = \frac{9}{8} = 1\frac{1}{8}$

Answer

Kai left one $\frac{3}{8}$ -inch seed off the plot, and Mia read the scale in fourths when it goes up in eighths. With both mistakes fixed, the three shortest seeds have a total length of $1\frac{1}{8}$ inches, not $1\frac{1}{2}$ inches.

The big idea to keep: every mark on a line plot is one measurement, and a stack is that measurement repeated once for each mark.

To find a total, multiply each value by its number of marks, then add.
To share equally, divide the total by the number of marks.
To check line plot work, make sure every measurement is on the plot, the scale is read one space at a time, and every mark is used.