Grade 2 · 2.MD.D.10 · 3.MD.B.3

Reading a bar graph

A bar on a graph has no pictures in it to count. How can you tell what number it stands for?

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Making bar graphsGrade 2 Reading a histogramMath 6 Association in a two way table and segmented bar graphsMath 8

Where you are 4 in Bar graphs and pictographs

Statistics7

Data displays12

Bar graphs and pictographs4

Picture graphs and their key

Reading a bar graph

Making bar graphs Scaled picture graphs

When you reach for it

Ekow’s class voted for a favorite season, and the teacher drew the votes as a bar graph. There is one bar for each season. A longer bar means more votes, so the winner can be seen at a glance. But a bar is a solid block. There are no pictures inside it to count, the way there are in a picture graph. So how does a reader find out that the Summer bar stands for 9 students and not 8 or 10?

The answer is the scale: a row of numbers along one edge of the graph, starting at 0 and going up in equal steps. A bar graph is a picture graph whose pictures have been pushed together into one bar, with the counting moved out to the edge. The bar shows how far. The scale says how many that is. Reading a bar graph means carrying the end of each bar over to the scale and reading the number there.

You reach for this skill whenever a bar graph is in front of you: a class vote, a weather chart, a count of anything sorted into a few kinds. Every bar graph in this lesson has at most four bars. Making one is a separate skill, Making bar graphs.

The lesson takes three days. Day 1 names the parts of a bar graph and reads bars on a scale that counts by 1, with bars drawn up the page and across it. Day 2 reads a scale that counts by 2, including a bar that stops between two labeled lines. Day 3 asks the graph questions (how many in all, how many more, how many fewer) and looks at what a graph cannot tell you.

The procedure

Day 1: the parts of a bar graph, and following a bar to the scale

Figure 1Ekow’s class season vote. The dashed line follows the top of the Summer bar straight across to the scale.

A bar graph has four parts to read, and they are read before any bar is.

The title, at the top, says what was counted: “Our Favorite Season”.
The labels under the bars name the categories: Spring, Summer, Fall, Winter. Each bar is one category.
The scale is the row of numbers up the side: 0, 1, 2, 3 and so on up to 10.
The scale label says what those numbers count: “Number of students”.

Read the title and the two kinds of labels first because they say what the numbers mean. Without them, a bar that reaches 9 could be 9 students, 9 days or 9 apples. Then look at the scale and find what it counts by. Take two numbers next to each other and find the jump from one to the next: from 3 to 4 is 1, from 4 to 5 is 1. On this graph the scale counts by 1, so each step up the side is 1 more student.

Now read one bar. Find where the bar ends, and go straight across from that end to the scale. The dashed line in the figure does this for the Summer bar. It leaves the top of the bar, runs level across the page, and meets the scale at 9. So 9 students chose summer.

Why straight across, and not on a slant? Each number on the scale marks one height above the bottom line. Everything level with the 9 is the same height above that line as the 9 is. The top of the Summer bar is level with the 9, so the bar is 9 steps tall. A slanted path would drift up or down on the way and land on a different height.

Why does the count come out right? Every bar stands on the same bottom line, and that line is the 0 of the scale. So the scale works like a ruler laid against the bar with its 0 at the bar’s foot. A bar 9 steps tall stands for 9, just as a pencil that reaches the 9 on a ruler laid from its end is 9 units long. It is also the picture graph again: if the Summer bar were cut at each step of the scale, it would fall into 9 equal blocks, one for each student.

Bars can also be drawn across the page. Then the categories are written down the left side, each bar grows to the right from a line on the left, and the scale runs along the bottom.

Figure 2Ramunas’s class drew its season vote with the bars going across. The scale runs along the bottom.

Everything else is the same. Read the title and labels, find what the scale counts by, then go from the end of a bar straight down to the scale. The light lines that run up from each number help the eye keep straight. Up or across, a bar graph reads the same way: the bar’s length, measured from the 0 line, is the count.

Day 2: a scale that counts by 2

Figure 3Puanani’s rain chart. The scale counts by 2, so the labeled lines are 0, 2, 4, 6 and so on.

Puanani keeps a rain chart for her class. Each month she counts the rainy days, and at the end she draws a bar graph. Counts like 13 would need a tall graph if every number had its own line, so her scale counts by 2: the labeled lines are 0, 2, 4, 6, 8, 10, 12, 14. To check, take two labels next to each other: from 6 to 8 is a jump of 2. Reading up the scale is skip counting by twos.

A bar that ends on a labeled line is read just as on Day 1: go straight across and read the number. March ends on the line for 10, so March had 10 rainy days.

A bar that ends between two labeled lines needs one more thought. April’s bar goes past the line for 12 but stops short of the line for 14, right in the middle. The space between those two lines is a piece of a number line from 12 to 14 (Number lines with different scales). That piece is 2 long, and the numbers in it are 12, 13 and 14. Halfway along a jump of 2 is 1 more than where it starts, so the middle of the space is 12 + 1 = 13. April had 13 rainy days.

Figure 4The scale from 10 to 16 laid on its side. Between two labeled lines sits one whole number, halfway.

Here is why the middle is the only place such a bar can stop. A count of rainy days is a whole number. Between 12 and 14 there is just one whole number, 13, and it sits halfway. So on a scale that counts by 2, a count that is not on a labeled line must be the number halfway between the two lines on either side of it. Those in-between numbers are the odd ones (1, 3, 5, 7 and so on), because the labeled lines are the even ones.

Later, scales count by 5 or by 10. There a bar that stops between two lines could stand for several different whole numbers, and reading it closely takes more care (Reading a graph: titles, labels and units). The idea is the same: find what one space on the scale is worth, then see how far into the space the bar goes.

Day 3: questions the graph can answer, and questions it cannot

Once each bar has been read, the graph holds a few numbers, and questions about those numbers are story problems. They come in the same three kinds as any adding and subtracting story.

Put together. “How many students voted in all?” adds every bar. “How many chose summer or winter?” adds two bars. Each vote is in exactly one bar, so the bars together are all the votes.

Take apart. “Of all the students who voted, how many did not choose summer?” starts from the whole and takes one part away. Adding the other bars gives the same number, which makes a good check.

Compare. “How many more rainy days in April than in June?” asks for the gap between two bars. Subtracting the smaller count from the larger one finds it (Subtraction as the difference). The graph shows the same gap as a picture: the taller bar sticks out past the shorter one, and the part that sticks out runs from the shorter bar’s number to the taller bar’s number on the scale. “How many fewer” is the same gap, asked from the other end. More of these question shapes are in How many more and how many fewer.

Some questions take two steps, because the number they need is not on any one bar: “Were there more rainy days in March and April together, or in May and June together?” First make the two totals, then compare them (Two-step addition and subtraction problems).

What a graph cannot tell you. A bar graph holds only the counts that were collected. The season graph says how many students in one class chose each season. It does not say which student chose what, why they chose it, what another class would choose, or which season is warmest. A question whose answer was never counted cannot be read off the graph, however carefully the bars are drawn. Before answering, ask: is this a question about the counts on these bars?

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 and 4 with Day 2, and Examples 5 to 7 with Day 3.

Example 1

Day 1: reading Ekow’s season vote

Use Ekow’s graph, “Our Favorite Season”, above. What does the graph count, and what does its scale count by? How many students chose each season? Which season had the most votes, and which had the fewest?

Solution

The title and the scale label say the graph counts students and the season each one chose. The scale reads 0, 1, 2, 3, and next-door numbers differ by 1, so the scale counts by 1.

Go from the top of each bar straight across to the scale. Spring’s top is level with 5, so 5 students chose spring. Summer’s top is level with 9 (the dashed line), so 9 students. Fall’s is level with 6, so 6 students. Winter’s is level with 3, so 3 students.

The biggest of 5, 9, 6 and 3 is 9, so summer had the most votes. The smallest is 3, so winter had the fewest. The picture agrees without any numbers: Summer is the tallest bar and Winter the shortest, because all four bars stand on the same 0 line.

Example 2

Day 1: bars that go across

Ramunas’s class, in Room 14, drew its season vote with the bars going across. How many students chose each season? Which season won in Room 14? Put the seasons in order from fewest votes to most.

Figure 5Ramunas’s class drew its season vote with the bars going across. The scale runs along the bottom.

Solution

The scale is along the bottom and reads 0, 1, 2, 3 up to 10, so it counts by 1. The bars grow to the right from the line on the left, so go from the right end of each bar straight down to the scale.

Spring ends above 7, so 7 students. Summer ends above 6, so 6 students. Fall ends above 4, so 4 students. Winter ends above 8, so 8 students.

The longest bar is Winter, 8, so winter won in Room 14. From fewest to most:

4 < 6 < 7 < 8

which is fall, summer, spring, winter. Each bar in that order reaches a little further right than the one before.

Notice that summer won in Ekow’s class, with 9 votes, but came second to last in Room 14. Each graph reports only its own class.

Example 3

Day 2: reading the rain chart on its lines

Use Puanani’s rain chart above. What does the scale count by? How many rainy days were there in March, and how many in May?

Solution

The labeled lines read 0, 2, 4, 6, 8, 10, 12, 14. From one label to the next is a jump of 2 (for example, 8 + 2 = 10), so the scale counts by 2. Counting up the lines is skip counting by twos: two, four, six, eight, ten.

The March bar’s top sits exactly on a light line. Follow that line straight left to the scale: it is the line for 10. March had 10 rainy days.

The May bar’s top sits on the line two below March’s, the line for 8. May had 8 rainy days.

Check with the picture: March’s bar is one space taller than May’s, and one space on this scale is 2. Indeed 8 + 2 = 10.

Example 4

Day 2: bars that stop between lines

On the same rain chart, how many rainy days were there in April, and how many in June?

Solution

The April bar goes past the line for 12 and stops before the line for 14, halfway between them. On this scale one space is 2, so halfway into a space is 1:

12 + 1 = 13

April had 13 rainy days. It has to be 13: a number of days is a whole number, and 13 is the only whole number between 12 and 14.

The June bar goes past the line for 4 and stops halfway to the line for 6. Halfway is 1 more than 4:

4 + 1 = 5

June had 5 rainy days. Check from the other side: 1 less than 6 is also 5, so the bar is truly in the middle.

Example 5

Day 3: in all, and taking apart

Use Ekow’s season graph. (a) How many students voted in all? (b) How many chose summer or winter? (c) How many students did not choose summer?

Solution

From Example 1 the counts are spring 5, summer 9, fall 6 and winter 3.

(a) Every student has exactly one vote, in exactly one bar, so adding the four bars gives everyone. Adding in a handy order is allowed, because order does not change a sum (the commutative property) and neither does which pair is added first (the associative property):

\begin{aligned} 5 + 3 &= 8 \\ 9 + 6 &= 15 \\ 15 + 8 &= 23 \end{aligned}

So 23 students voted.

(b) Summer or winter puts two bars together: 9 + 3 = 12 students.

23 - 9 = 14

and 14 students did not choose summer. Check by adding the other three bars: 5 + 6 + 3 = 14.

Example 6

Day 3: how many more, how many fewer

Use Puanani’s rain chart. (a) How many more rainy days were there in April than in June? (b) How many fewer rainy days were there in May than in March? (c) Were there more rainy days in March and April together, or in May and June together, and by how many?

Solution

From Examples 3 and 4 the counts are March 10, April 13, May 8 and June 5.

(a) April has 13 and June has 5. The April bar sticks out above the June bar from 5 up to 13 on the scale. The length of that piece is the gap:

13 - 5 = 8

April had 8 more rainy days than June. Check by adding back: 5 + 8 = 13.

(b) March has 10 and May has 8. May’s bar stops one space below March’s, and one space is 2:

10 - 8 = 2

May had 2 fewer rainy days than March.

\begin{aligned} 10 + 13 &= 23 \\ 8 + 5 &= 13 \end{aligned}

March and April had more. To find how many more, compare the two totals:

23 - 13 = 10

There were 10 more rainy days in March and April together than in May and June together.

Example 7

Day 3: what the graph can and cannot tell

Here are five questions about Ramunas’s Room 14 graph. Which can the graph answer? Answer those, and say why the others cannot be answered from it.

How many students in Room 14 voted?
Did Ramunas choose winter?
Which season is the coldest?
If Room 14 voted again next year, would winter win?
Did more students choose spring and summer together than fall and winter together?

Solution

From Example 2 the counts are spring 7, summer 6, fall 4 and winter 8.

Question 1 is about the counts, so the graph answers it. Add the four bars, making a ten first: 6 + 4 = 10, then 7 + 8 = 15, and 10 + 15 = 25. So 25 students voted.

Question 5 is about the counts too, in two steps. Spring and summer: 7 + 6 = 13. Fall and winter: 4 + 8 = 12. Since 13 > 12, yes, by 13 - 12 = 1 vote.

Question 2 cannot be answered. The Winter bar says 8 students chose winter, but not which 8. Each vote went into a bar without a name on it.

Question 3 cannot be answered. The graph counts which season students like, not how cold any season is. Winter got the most votes, and that says nothing about temperature.

Question 4 cannot be answered. The graph shows one vote, taken once. Next year’s class could choose differently, and nothing on this graph has been counted about next year.

Boundaries

Read the scale before reading a bar. The same bar height means a different number on a scale that counts by 1 and on one that counts by 2. Every reading in this lesson starts by finding what one space on the scale is worth.
The scale has to start at 0, and the bars have to start at the 0 line. The count is the length from 0. A bar that starts somewhere else, or a scale whose first number is not 0, breaks the ruler: the lengths no longer compare fairly, and a bar twice as long need not stand for twice as many.
The steps of the scale have to be equal. Reading halfway between 12 and 14 as 13 only works because every space on the scale is worth the same 2.
A bar between two lines is exact only when one whole number fits there. On a scale that counts by 2, the only whole number between two neighboring labels is the one halfway, so the reading is exact. On a scale that counts by 5 or 10 several numbers fit, and a bar between lines can only be read as “about” (Reading a graph: titles, labels and units).
Each thing counted sits in exactly one bar. That is why adding the bars gives the total. If one student could vote twice, the bars together would be more than the number of students.
A graph answers questions about its counts and nothing else. Who, why, another group, another time, or anything that was not counted lies outside it.

Where this goes next

Drawing a bar graph from a tally or a table is Making bar graphs: choosing the scale, labeling it, and making each bar end at the right height. Every check made here when reading (a title, labels, a scale that starts at 0 in equal steps) becomes a choice the maker has to get right.

A dot plot, or line plot, also counts things, but its categories are numbers on a number line, and it shows one mark for each thing instead of one bar for each category (Reading dot plots: clusters, gaps and peaks, Making a dot plot).

Later the scale counts by 5, 10 or more, so a bar between lines takes estimating, and a picture graph’s key may say one picture stands for several things (Scaled picture graphs, Reading a graph: titles, labels and units). Further on, a histogram is a bar graph whose bars stand for ranges of numbers, read with the same move: from the end of the bar, straight across to the scale.

Exercises

Day 1: the parts of a bar graph, and following a bar to the scale

Exercises 1 to 5 use this graph of Puanani’s rain chart for the end of the year.

Figure 6The graph for Exercises 1 to 5, 13 to 16 and 22.

What does this graph count? What does its scale count by?
How many rainy days were there in October?
How many rainy days were there in November?
Which month had the fewest rainy days, and how many?
Put the four months in order from fewest rainy days to most.
A bar graph’s scale counts by 1. A bar ends level with the line just below the line labeled 15. What number does the bar stand for?

Day 2: a scale that counts by 2

Exercises 7 to 10 use this graph of a season vote taken in three classes together.

Figure 7The graph for Exercises 7 to 10, 17 to 21 and 24.

How many students chose summer?
How many students chose winter?
How many students chose spring?
How many students chose fall?
A bar graph’s scale counts by 2. A bar stops halfway between the lines labeled 16 and 18. What number does it stand for?
On a scale that counts by 2, a bar stands for 11. Between which two labeled lines does it stop?

Day 3: questions the graph can answer

Exercises 13 to 16 use the rain chart for September to December. Exercises 17 to 21 use the season vote of three classes together.

How many rainy days were there in all four months?
How many rainy days were there in October and November together?
How many more rainy days were there in November than in September?
How many fewer rainy days were there in December than in October?
How many students voted in all?
How many more students chose summer than fall?
How many students did not choose winter?
Did more students choose spring and fall together, or summer alone? By how many?
Four more students vote, and all four choose winter. How many students chose winter now? Is that more or fewer than fall, and by how many?
Puanani also counted rainy days in August, but that bar is missing from her graph. She knows there were 40 rainy days from August to December in all. How many rainy days were there in August?

Explain

Why must every bar in a bar graph start from the same 0 line?
Can the season graph for three classes tell you which season is the warmest? Explain.
On a scale that counts by 2, a bar showing a number of rainy days stops between the lines for 6 and 8. Why must it stop exactly halfway, and what number does it show?

Answers

It counts rainy days in each month from September to December; the title and the scale label both say so. The scale reads 0, 1, 2, 3, so it counts by 1.
9. The October bar ends above 9 on the scale.
11. The November bar ends above 11.
September, with 6 rainy days. Its bar is the shortest.
September (6), December (7), October (9), November (11), because 6 < 7 < 9 < 11.
14. On a scale that counts by 1, the line just below 15 is 15 - 1 = 14.
18. The Summer bar ends on the line for 18.
6. The Winter bar ends on the line for 6.
15. The Spring bar stops halfway between 14 and 16, and 14 + 1 = 15.
9. The Fall bar stops halfway between 8 and 10, and 8 + 1 = 9.
17. Halfway into a space of 2 is 1, and 16 + 1 = 17.
Between 10 and 12. The labeled lines are the even numbers, and 11 is 1 more than 10 and 1 less than 12.
33. Make a ten first: 9 + 11 = 20, 6 + 7 = 13, and 20 + 13 = 33.
20. Put the two bars together: 9 + 11 = 20.
5 more. 11 - 6 = 5. Check: 6 + 5 = 11.
2 fewer. 9 - 7 = 2.
48. 15 + 18 = 33, 9 + 6 = 15, and 33 + 15 = 48.
9 more. 18 - 9 = 9.
42. Take the winter part from the whole: 48 - 6 = 42. Check: 15 + 18 + 9 = 42.
Spring and fall together, by 6. First make the total: 15 + 9 = 24. Then compare: 24 - 18 = 6.
10 students chose winter now, since 6 + 4 = 10. That is 1 more than fall, since 10 - 9 = 1.
7. The four bars on the graph make 33 (Exercise 13), and the whole is 40, so August is the missing part: 40 - 33 = 7. Check: 33 + 7 = 40.
The count is the length of the bar measured from 0, the way a ruler measures from its 0. If one bar started higher up, it would look shorter than its count, and comparing lengths would give the wrong answer. Starting every bar at the same 0 line keeps a longer bar meaning a bigger count.
No. The graph counts how many students like each season best. It counts nothing about temperature, so a question about warmth is not a question about its counts. Summer got the most votes, but that is about what students like, not about how warm it is.
A number of rainy days is a whole number. The only whole number between 6 and 8 is 7, and 7 is 1 more than 6 and 1 less than 8, which is exactly halfway. So the bar stops halfway and shows 7.