Grade 3 · 2.MD.D.10 · N.Q.A.1

Reading a graph: titles, labels and units

Someone hands you a bar graph you have never seen. Before you read a single bar, what do you need to know to be sure what it says?

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Making bar graphsGrade 2

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Choosing a display for a questionMath 6 Misleading graphsMath 6

Where you are 12 in Data displays

Statistics7

Data displays12

▴ 2 above Line graphs Histograms4

Reading a graph: titles, labels and units

Scatter plots2 Stem and leaf plots ▾ Show all 12 (5 more)

When you reach for it

Most graphs you meet were drawn by somebody else: a poster in the hallway, a page in a science book, a chart a classmate made after a survey. The person who drew it knew what it was about. You only have what is printed on the page.

Making a graph is covered in Making bar graphs: a table of counts becomes one bar per category, each bar as tall as its count on a numbered scale. Reading a graph runs that the other way, and it starts with the words, not the bars. A bar on its own is just a shaded box. It becomes a number only once you know what the numbers on the side count by, and it becomes a fact only once you know what was counted.

This lesson takes two class days. Part 1 reads the frame of a bar graph and then its values, including a bar that stops between two marks. Part 2 looks at the same data drawn on two different scales, at picture graphs and their keys, and at graphs with words missing.

The procedure

Part 1: the frame first, then the values

Every bar graph has a frame of words and numbers around the bars. There are four parts to it.

The title says what was counted, and often where or when. “Students in Each Field Day Event” tells you the bars count students and that each bar is one event.
The category labels under the bars name what each bar stands for: one event, one person, one job.
The label on the value axis (the side with the numbers) says what the numbers count, and in what unit: students, laps, minutes, bottles.
The scale is the list of numbers up that side. What matters most is how much one space between two marks stands for.

Read the scale by taking two numbers next to each other and finding the gap. Below, the marks say 0, 5, 10, and 5 - 0 = 5, 10 - 5 = 5. Check one more pair higher up, 30 - 25 = 5, to be sure the spaces are all the same. This scale counts by 5, so each space stands for 5 students.

Figure 1The title, the labels and the scale come first. Only then the bars.

Now read a value. Run your eye from the top of a bar straight across to the scale. If the top sits on a mark, the value is that mark’s number. The Sack race bar stops on the mark for 25, so 25 students took part.

A bar can stop between two marks. If it stops exactly halfway, the value is halfway between the two numbers. Halfway across a space of 10 is 5, so halfway between 20 and 30 is 25. Halfway across a space of 2 is 1, so halfway between 12 and 14 is 13. Both of those are whole numbers, and a count can be read exactly.

A space of 5 is different. Halfway between 15 and 20 is 17 and a half, and nobody counts half a student. A bar of students that stops near halfway between 15 and 20 stands for 17 or 18, and the picture is not fine enough to say which. The honest reading is “about 17 or 18”, or “between 15 and 20”. That is not a failure of the reader; it is all the graph shows.

Once the values are read, the questions a graph answers are the ones a table answers. “How many more” is a subtraction, as in Subtraction as the difference, and “how many in all” is an addition.

Part 2: two scales, keys, and missing words

The scale is chosen by whoever draws the graph. The same counts can go on a scale by 2 or a scale by 10. The bars are the same heights compared with each other, because both scales start at 0 and space their marks evenly. What changes is how many marks there are to read against. A scale by 2 has a mark every 2, so most counts sit on a mark or exactly halfway. A scale by 10 has few marks, so a count like 13 falls inside a big space and can only be read roughly. A graph with fewer marks often fits in less room; a graph with more marks gives exact counts. Both are honest.

A picture graph shows each count as a row of little pictures. The key tells how many each picture stands for, so the key does the scale’s job. A row of 6 pictures is 6 things if each picture is 1, and 60 things if each picture is 10. When one picture stands for several, a row can end in part of a picture; that case is the subject of Scaled picture graphs.

When a word is missing, some questions cannot be answered. Each part of the frame answers its own question.

Missing	What you can no longer say
title	what was counted, or where and when
value-axis label	the unit: laps, minutes, students
numbers on the scale	any value at all, only which bar is taller
key on a picture graph	how many each picture stands for

Without the scale numbers you can still say which bar is tallest. Without the title you can still read every number. But a number with no idea of what it counts does not answer a question about the world.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 to 7 with Part 2.

Example 1

Reading the field day graph

Use the field day graph above. What does the graph show, and what does each space on the scale stand for? How many students were in the Sack race, the Tug-of-war and the Balloon toss? How many more students were in the Tug-of-war than in the Balloon toss?

Solution

The title says the bars count students in each field day event, and the value-axis label agrees: “Number of students”. There are four bars, one per event.

The marks read 0, 5, 10, 15, 20, 25, 30. Next to each other they differ by 5, so each space stands for 5 students.

The Sack race bar stops on the mark for 25, so 25 students. The Tug-of-war bar stops on the top mark, 30, so 30 students. The Balloon toss bar stops on the mark for 10, so 10 students.

“How many more” compares two counts, so subtract:

30 - 10 = 20

There were 20 more students in the Tug-of-war than in the Balloon toss. Check by adding back: 10 + 20 = 30.

Example 2

The bar that stops between marks

On the same graph, how many students were in the Three-legged race? Were there more than 15? How many students took part in all four events together?

Solution

The Three-legged race bar stops above the mark for 15 and below the mark for 20, close to halfway. One space is 5 students, and halfway across it is 2 and a half, so exactly halfway would be 17 and a half students. A count of students is a whole number, so the bar stands for 17 or 18. The graph cannot say which. The reading is about 17 or 18 students.

Even so, one question has a sure answer. The bar is plainly above the 15 mark, so more than 15 students were in the Three-legged race.

For the total, add the three bars that sit on marks first:

25 + 30 + 10 = 65

Then the Three-legged race adds 17 or 18: 65 + 17 = 82 and 65 + 18 = 83. So 82 or 83 students took part in all, and the graph cannot narrow it further. The person who drew the graph knows; the reader can only give the two possible totals.

Example 3

A scale with only two numbers

A field day graph shows the number of students at the Jump rope station. Its value axis has 0 at the bottom and 40 at the top line, and 8 equal spaces between them, with no other numbers printed. The Jump rope bar stops on the sixth line above 0. How many students jumped rope?

Solution

The scale is not printed, so work it out. From 0 to 40 is 8 equal spaces, so each space is one eighth of 40:

40 \div 8 = 5

Each space stands for 5 students. Check: 8 \times 5 = 40, which lands on the top line. The bar stops 6 spaces above 0, so it stands for

6 \times 5 = 30

students. Counting up by 5s along the lines gives the same thing: 5, 10, 15, 20, 25, 30.

Example 4

The pet log on a scale by 2

Ningali, Haunani and Fetu keep the class pet log for the guinea pig. Their graph of October’s jobs is below. Read each bar. How many more times was the guinea pig fed than its cage cleaned? How many jobs were done in all?

Figure 2The pet log on a scale that counts by 2.

Solution

The title says the bars count jobs done in October, and the value axis is “Number of times”. The marks read 0, 2, 4, \ldots, 20, so each space stands for 2 times.

Feeding stops on the mark for 18. Water stops exactly halfway between 12 and 14; half a space is 1, so Water is 13. Cleaning stops on 6, and Weighing on 4.

18 - 6 = 12

The guinea pig was fed 12 more times than its cage was cleaned. In all:

18 + 13 + 6 + 4 = 41

So 41 jobs were done in October.

Example 5

The same pet log on a scale by 10

Here are the same counts drawn on a scale that counts by 10. What can still be read from this graph, and what cannot?

Figure 3The same pet log on a scale that counts by 10. Same bars, fewer marks to read them against.

Solution

The only marks are 0, 10 and 20, so each space stands for 10 times. Read each bar against those marks.

Feeding stops a little below 20: more than 10, under 20, and close to the top. Water stops a little above 10. Halfway between 10 and 20 is 15, and the Water bar is below halfway, so Water is between 10 and 15. Cleaning stops a little above halfway between 0 and 10, so it is a bit more than 5. Weighing stops below halfway, so it is less than 5.

What can still be said for sure: Feeding happened most and Weighing least, and the order of the bars from tallest to shortest is the same as on the scale by 2. Both graphs start at 0, so the bars keep the same heights compared with each other.

What cannot be read is any exact count. This graph cannot answer “how many more times was the guinea pig fed than its cage cleaned?” with a number; the scale by 2 answered 12. The data are the same. The scale by 10 is shorter and simpler, and it shows the pattern; the scale by 2 is finer, and it shows the counts.

Example 6

A graph with no title

Use the graph with no title below. Can you tell how many laps Haunani ran? What would the graph need before that question has an answer, and what is the answer then?

Figure 4Every bar can be read, but nothing says what is being counted.

Solution

The numbers can all be read. The marks go up by 2, Ningali’s bar stops on 6, Haunani’s on 10, and Fetu’s on 4.

But nothing on the graph says what the bars count. The 10 could be 10 laps, 10 minutes, 10 books, or 10 ribbons. So the question “how many laps did Haunani run?” has no answer from this graph. Even “who ran the most laps?” has no answer, since the bars may not be laps at all. The only safe statement is that Haunani’s bar is the tallest.

Suppose the person who drew it adds the title “Laps Run at Field Day” and the value-axis label “Number of laps”. Now Haunani’s bar says 10 laps, and questions about laps have answers. For example, Haunani ran 10 - 4 = 6 more laps than Fetu.

Example 7

Reading a picture graph’s key

Use the picture graph of field day points. How many points did each team win? How many more points did the Blue team win than the Green team? How many points were won in all?

Figure 5In a picture graph the key does the scale’s job.

Solution

The title says the rows show points won at field day. The key says each circle stands for 10 points, so each row is a count of tens.

Team	Circles	Points
Red	4	4 \times 10 = 40
Blue	6	6 \times 10 = 60
Green	3	3 \times 10 = 30
Gold	5	5 \times 10 = 50

The Blue team won 60 - 30 = 30 more points than the Green team. Counting circles gives the same answer a second way: Blue has 3 more circles, and 3 \times 10 = 30.

In all, 40 + 60 + 30 + 50 = 180 points. Or count every circle first, 4 + 6 + 3 + 5 = 18 circles, and 18 tens is 180.

Cover the key with your hand and try again. You can still say Blue has the most circles and 3 more than Green. But “how many points?” now has no answer: the circles could be 1 point each, or 5, or 10.

Boundaries

This lesson reads graphs of categories: events, jobs, people, teams. A graph whose bottom axis is a number or a time, read as a line, is Line graphs.
A value can be read only as exactly as the scale allows. A bar on a mark, or exactly halfway on a scale by 2 or by 10, gives an exact whole count. A bar between marks on a scale by 5, or anywhere inside a big space, gives a range.
Reading a bar against the scale assumes the scale starts at 0 and its spaces are equal. A graph that breaks either rule can make a difference look bigger or smaller than it is; that is Misleading graphs.
A graph shows only what was counted. A graph of this year’s field day events says nothing about next year’s.
A picture graph with no key, or a bar graph with no numbers on its scale, gives only which category is bigger, never how many.

Where this goes next

The same move, find what one space stands for and then read between the marks, is how a ruler, a measuring cup or a thermometer is read; see Measuring tools and reading their scales. Picture graphs whose rows end in half a picture are read in Scaled picture graphs, and the key itself is Picture graphs and their key. Reading a bar’s count against its scale is Reading a bar graph.

Later, the frame grows. The title and labels become the individuals and variables of Individuals, variables and units, and the choice of which kind of graph to draw becomes Choosing a display for a question. When a scale does not start at 0, or pictures grow in two directions at once, Misleading graphs shows how the picture can mislead even when every number is right.

Exercises

Part 1: reading a bar graph

Use the graph of water bottles handed out at field day.

Figure 6The graph for Exercises 1 to 7.

What does the graph count, and what does each bar stand for?
What does one space on the scale stand for?
How many bottles were handed out at the Snack table?
How many bottles were handed out at the Finish line?
How many more bottles were handed out at the Snack table than at the Shade tent?
How many bottles were handed out in all?
Put the Start line and Shade tent together. Did they hand out more bottles than the Finish line, or fewer, and by how many?

Part 1: scales and marks

A scale counts by 2. A bar stops exactly halfway between 8 and 10. What number does the bar stand for?
A scale counts by 10. A bar stops exactly halfway between 70 and 80. What number does it stand for?
A scale counts by 5, and the bars count children. A bar stops close to halfway between 25 and 30. What can you say about the number of children?
A value axis has 0 at the bottom and 18 at the top line, with 9 equal spaces and no other numbers. What does one space stand for? A bar stops on the seventh line above 0; what does it stand for?

Part 2: keys and two scales

Use the pet log picture graph for Exercises 12 to 15.

Figure 7The graph for Exercises 12 to 15.

How many days did Ningali care for the guinea pig?
How many more days did Fetu care for the guinea pig than Haunani?
How many days of care are shown in all?
Suppose the key said each circle stands for 5 days instead. How many days would Ningali’s row show?
In November the pet log counts are: Feeding 20, Water 16, Cleaning 8, Weighing 5. On a scale that counts by 2, which bars stop on a mark and which stop exactly halfway between two marks?
The same November counts are drawn on a scale that counts by 10. Which bars can still be read exactly, and what can you say about the others?

Part 2: missing words

A bar graph has bars labeled Ningali, Haunani and Fetu and a scale that counts by 5, but no title and no value-axis label. Ningali’s bar stops on 10, Haunani’s on 15 and Fetu’s on 5. Which of these can you answer: (a) whose bar is tallest? (b) how many books did Haunani read?
A picture graph has rows for four field day events with 3, 5, 2 and 4 pictures, and no key. How many more pictures does the longest row have than the shortest? Can you say how many more students?
A graph is titled “Time Spent at Each Field Day Event” and its value axis is labeled “Number of minutes”. One bar stops on 45. What does the 45 tell you?
A friend draws the pet log on a scale by 10 to fit a small poster. Another wants to know exactly how many times the cage was cleaned. Which scale would you suggest for each, and why?

Explaining

Why read the scale before reading any bar? Give a bar whose value comes out wrong if you assume the scale counts by 1.
Within one graph whose scale starts at 0, a taller bar always stands for more. Is that still true when you compare a bar on one graph with a bar on a different graph? Give an example.
Why can a bar counting students never stand for exactly the halfway point between 15 and 20 on a scale by 5, while a bar halfway between 12 and 14 on a scale by 2 can be read exactly?
A bar graph has its category labels and its scale, but its title has been torn off. Write one question it can still answer and one it cannot, and say why.

Answers

It counts water bottles handed out at field day. Each bar is one station: Start line, Shade tent, Snack table and Finish line. The title and the label “Number of bottles” say so.
10 bottles. Next to each other the marks differ by 10 - 0 = 10.
40 bottles. The Snack table bar stops on the mark for 40.
55 bottles. The bar stops exactly halfway between 50 and 60, and half of a space of 10 is 5.
20 more. The Snack table is 40 and the Shade tent is 20, and 40 - 20 = 20.
145 bottles. 30 + 20 + 40 + 55 = 145.
Fewer, by 5. Together they handed out 30 + 20 = 50, and the Finish line handed out 55, so 55 - 50 = 5.
9. Half of a space of 2 is 1, and 8 + 1 = 9.
75. Half of a space of 10 is 5, and 70 + 5 = 75.
There were about 27 or 28 children: more than 25, fewer than 30. Exactly halfway would be 27 and a half, which is not a count of children, so the graph cannot say which.
Each space stands for 2, since 18 \div 9 = 2. The bar stands for 14, since 7 \times 2 = 14.
10 days. Ningali’s row has 5 circles, and 5 \times 2 = 10.
2 more days. Fetu has 4 \times 2 = 8 and Haunani 3 \times 2 = 6, and 8 - 6 = 2.
24 days. 10 + 6 + 8 = 24, or 12 circles in all and 12 \times 2 = 24.
25 days. The same 5 circles, now 5 \times 5 = 25. The pictures did not change; the key did.
Feeding 20, Water 16 and Cleaning 8 are even, so they stop on marks. Weighing 5 stops exactly halfway between 4 and 6.
Feeding stops on the mark for 20 and can be read exactly. Weighing stops exactly halfway between 0 and 10, so it reads 5. Water is between 15 and 20, a little past halfway from 10 to 20, and Cleaning is between 5 and 10, a little below the 10 mark; neither can be read exactly.
(a) Yes: Haunani’s bar is tallest. (b) No: nothing says the bars count books, so the 15 could be 15 of anything.
3 more pictures, since 5 - 2 = 3. Without a key there is no way to say how many students that is: each picture could stand for 1, 2, 10 or any number.
45 minutes were spent at that event. The title says the bars measure time at each event, and the label gives the unit, minutes.
The scale by 10 suits the small poster, since it needs fewer marks and shows which job happened most. The scale by 2 suits the question about cleaning, since every count sits on a mark or exactly halfway and can be read exactly.
The same bar height means different numbers on different scales. A bar 3 spaces tall on a scale by 10 stands for 30; assuming each space is 1 would read it as 3.
No. On a graph that counts by 2, a bar 4 spaces tall stands for 8. On a graph that counts by 10, a bar 2 spaces tall stands for 20. The shorter bar stands for more, because the two scales are different.
Halfway across a space of 5 is 2 and a half, so the halfway point between 15 and 20 is 17 and a half, and a count of students cannot be a half. Halfway across a space of 2 is 1, so halfway between 12 and 14 is 13, a whole number.
For example, “which category has the biggest count?” or “how much bigger is one bar than another?” can still be answered, because the labels and scale are there. “What was counted?” or “how many students chose soccer?” cannot, because without the title nobody knows what the numbers count.