Grade 2 · 2.MD.D.10

Picture graphs and their key

Twenty children each voted for a favorite fruit. How can one page show every vote at once, so that anyone can see which fruit won, and by how much?

Unlocks

Scaled picture graphsGrade 3

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

Why this exists

Vladislava’s class votes for a favorite fruit. Each child says one fruit, and the teacher writes it down. At the end there is a list of twenty words: banana, grapes, strawberry, banana, and so on. Every vote is on that list, but nobody can look at it and say which fruit won. To answer, you have to sort and count the whole list again.

A picture graph (also called a pictograph) does the sorting and counting once, on paper, so that it never has to be done again. Each fruit gets its own row, and each vote becomes one small picture in its fruit’s row. Now the answer can be seen. The longest row won, and the rows can be compared side by side.

The graph also carries a small note called the key. It says what one picture stands for. In this lesson the key always says the same thing: one picture stands for one thing. That might seem too obvious to write. It is not. The key is the promise that lets a reader trust the count: it tells them that counting the pictures in a row gives the number of votes, and not some other number. A reader who meets a graph they did not make reads the key first, every time.

This lesson takes three days. Day 1 reads a picture graph. Day 2 makes one from a tally or a list. Day 3 asks the graph questions: how many in all, how many more, how many fewer. Each graph here has at most four categories.

Definition

A picture graph shows how many things fall into each of a few groups, called categories, using one picture for each thing counted. It has:

a title, which says what was counted;
a name for each category, written at the start of its row (or under its column);
pictures, one for each thing counted, placed in the row of its category;
a key, which says what one picture stands for, such as “each circle stands for 1 student”.

All the pictures in one graph are the same shape and the same size, and every row starts at the same line. A graph may run in rows across the page or in columns up the page; the rules are the same either way.

The key is the note that says how many things one picture stands for. Every key in this lesson says that one picture stands for 1 thing.

Intuition

Day 1: read the key, then the rows

Figure 1Vladislava’s class fruit vote. The key says each circle stands for 1 student.

Start with the title: “Our favorite fruit”. It says what the graph is about. Then read the key: each circle stands for 1 student. Only now do the pictures mean anything. Each circle in the bananas row is one student who chose bananas.

So the number of circles in a row is the number of students who chose that fruit. Count the bananas row: 1, 2, 3, 4, 5, 6. Six students chose bananas.

Because every circle is the same size and every row starts at the same line, the rows can be compared before any counting at all. The strawberries row reaches furthest, so strawberries had the most votes. The oranges row is the shortest, so oranges had the fewest.

A graph in columns reads the same way, from the bottom up. Each column is a stack of pictures standing on one line, and the tallest stack is the biggest count.

Figure 2Dobromira’s lost tooth chart. Each column grows up from the same line.

Day 2: make a graph that tells the truth

A picture graph is made from counts. The counts may come from a tally or a table, or from a plain list that still needs sorting. Either way, the graph is finished when every thing counted has exactly one picture, in the row of its category.

Two choices make the graph fair to read. Every picture is the same size, and every row starts at the same line, with the pictures evenly spaced. Then a longer row always means more things. Break either rule and the lengths start to lie:

Figure 3The same counts drawn twice. Only graph A lets the lengths tell the truth.

Both graphs show 6 ants and 4 bees. In graph A the ants row is longer, as it should be. In graph B the bee circles are bigger and spread out, so the bees row looks longer, even though there are still only 4 bees. A reader who trusts the lengths is fooled. Lined-up, same-size pictures are what let a glance at the graph give the right answer.

The maker also writes the title, the category names and the key. A category with nothing in it keeps its name and gets an empty row: the empty row says zero.

Day 3: questions the graph can answer

Once each row has been counted, the graph answers questions about the counts.

How many in all. Add the rows. Every thing counted has exactly one picture, so the total of the rows is the total number of things. In the fruit vote, that is the number of students who voted.

How many more, how many fewer. Find the gap between two rows. Line the two rows up: the pictures in the shorter row each have a partner in the longer row, and the pictures left without a partner are the gap. Subtracting the smaller count from the larger one gives the same number (Subtraction as the difference). “How many more” and “how many fewer” ask about the same gap from its two ends.

Two steps. Some questions need a number that is not on the graph yet, such as “how many chose bananas or grapes together?” The first step makes that number; the second step uses it.

Results

One picture, one thing. When the key says each picture stands for 1, the number of pictures in a row is the count for that category.

The pictures add up to the whole. The number of pictures in the whole graph is the number of things counted.

Longer means more. When the pictures are the same size and the rows start at the same line, the longer row has the bigger count, and the unmatched pictures at its end are how many more.

Why it is true

Each thing counted was given exactly one picture, in exactly one row. That is a matching: every thing has its picture, and every picture has its thing. Two groups matched one to one have the same number, so counting the pictures in a row is the same as counting the things in that category. The key is what tells the reader that the matching is one picture to one thing.

No thing was skipped and none was drawn twice, so all the pictures together match all the things together. Their numbers are the same.

When pictures are the same size and evenly spaced from the same line, each picture takes up the same length. A row of 6 pictures is then 6 of those lengths long, and a row of 4 is 4 of them, so the row with more pictures is longer. Lay the rows side by side and the first 4 pictures of each row sit one above the other, matched. What sticks out past the shorter row is the extra pictures, which is exactly how many more the longer row has.

Worked examples

Example 1

Day 1: reading the fruit vote

Use Vladislava’s fruit vote graph above. How many students chose each fruit? Which fruit had the most votes, and which had the fewest?

Solution

Read the key first: each circle stands for 1 student. So each count is the number of circles in the row.

Bananas has 6 circles, so 6 students. Grapes has 4 circles, so 4 students. Oranges has 3 circles, so 3 students. Strawberries has 7 circles, so 7 students.

The biggest of 6, 4, 3 and 7 is 7, so strawberries had the most votes. The smallest is 3, so oranges had the fewest. The rows say the same thing without the numbers: the strawberries row is the longest and the oranges row is the shortest, because every circle is the same size and every row starts at the same line.

Example 2

Day 1: reading columns

Use Dobromira’s lost tooth chart above. How many teeth did the class lose in October? Which month had the fewest lost teeth? Put the months in order from fewest teeth to most.

Solution

The key says each square stands for 1 tooth. The columns stand on a line and grow upward, so count each stack from the bottom.

October has 5 squares, so the class lost 5 teeth in October. September has 3, November has 2 and December has 6.

November’s column is the shortest, so November had the fewest lost teeth, 2. From fewest to most:

2 < 3 < 5 < 6

which is November, September, October, December. Check with the picture: each column in that order stands a little taller than the one before.

Example 3

Day 2: from a tally to a graph

Zhuldyz walks around the school garden and makes a tally mark for each bug she finds. Make a picture graph from her tally.

Figure 4Zhuldyz’s tally, made one mark at a time as she found each bug.

Solution

First turn each row of the tally into a number. A bundle of tally marks is 5.

Ladybugs is one bundle: 5. Ants is one bundle and 3 more: 5 + 3 = 8. Bees is 3 loose marks: 3. Butterflies is 4 loose marks: 4.

Now choose a picture, a small diamond, and write the key: each diamond stands for 1 bug. Draw a line, write the four bug names to its left, and put the right number of diamonds in each row: 5, 8, 3 and 4. Every diamond is the same size and every row starts at the line, evenly spaced. Add the title, “Bugs in the school garden”.

Figure 5The picture graph Zhuldyz makes from her tally.

Check: the tally has 5 + 8 + 3 + 4 = 20 marks, and the graph has 20 diamonds. One picture for each bug, none missed and none extra.

Example 4

Day 2: from a list, with an empty category

During a short school week, each time a child in Dobromira’s class loses a tooth, she writes the day on a list:

Monday, Wednesday, Monday, Thursday, Monday, Wednesday, Monday

Make a picture graph in columns for Monday, Tuesday, Wednesday and Thursday.

Solution

Sort the list first. Go along it once, and cross out each word as you count it, so that no word is counted twice and none is missed. Monday appears 4 times, Wednesday 2 times and Thursday 1 time. Tuesday does not appear at all, so its count is 0.

Check the sorting: 4 + 0 + 2 + 1 = 7, and the list has 7 words.

Draw a line across the page and write the four days under it. Above Monday stack 4 squares, above Wednesday 2 and above Thursday 1, all the same size. Write the key: each square stands for 1 tooth. Add the title.

Figure 6Tuesday keeps its label with no squares above it. That empty column says zero.

Tuesday keeps its name even though its column is empty. Leaving it out would hide the fact that the class looked on Tuesday and no teeth were lost. An empty column says zero.

Example 5

Day 3: how many in all

Use Vladislava’s fruit vote. How many students voted in all? How many chose bananas or grapes?

Solution

The counts from Example 1 are bananas 6, grapes 4, oranges 3 and strawberries 7. Each student has exactly one circle, so all the circles together are all the votes.

Add the four counts. Pairs that make ten are quickest (Making ten and bridging through ten): 6 and 4 make 10, and 3 and 7 make 10. So

\begin{aligned} 6 + 4 &= 10 \\ 3 + 7 &= 10 \\ 10 + 10 &= 20 \end{aligned}

and 20 students voted. Adding in a different order is allowed: addition does not care about order (the commutative property) or about which pair is added first (the associative property).

Bananas or grapes is two rows put together: 6 + 4 = 10 students.

Example 6

Day 3: how many more, how many fewer

Use Zhuldyz’s garden bug graph. (a) How many more ants than bees did she find? (b) How many fewer butterflies than ladybugs?

Solution

The counts from Example 3 are ladybugs 5, ants 8, bees 3 and butterflies 4.

(a) Line up the ants row and the bees row. The 3 bees each have an ant diamond above them. The ant diamonds left without a partner are the extra ants. Count them: 5. Subtracting says the same:

8 - 3 = 5

so Zhuldyz found 5 more ants than bees. Check by counting on from 3: “four, five, six, seven, eight” is 5 more.

(b) Ladybugs has 5 and butterflies has 4. The butterflies row stops one diamond short of the ladybugs row, so

5 - 4 = 1

and she found 1 fewer butterfly than ladybugs.

Example 7

Day 3: comparing two parts of the year

Use Dobromira’s lost tooth chart. Did the class lose more teeth in September and October together, or in November and December together? How many more teeth would November need to match December?

Solution

The counts from Example 2 are September 3, October 5, November 2 and December 6. No column shows “September and October”, so the first step is to make those two numbers.

\begin{aligned} 3 + 5 &= 8 \\ 2 + 6 &= 8 \end{aligned}

Both are 8. The class lost the same number of teeth in each pair of months, even though December alone had the tallest column and November alone the shortest. The big month and the small month balance out.

For the second question, November has 2 and December has 6. The question is 2 + \square = 6. Count on from 2: “three, four, five, six”, which is 4. So November would need 4 more squares to match December. Check: 6 - 2 = 4.

Boundaries

The key in this lesson always says one. Every count here is read by counting pictures, because each picture stands for 1 thing. A key that says one picture stands for 2, 5 or 10 things changes how a row is read; that is Scaled picture graphs. Whatever the key, it holds for every picture in the graph.
Each thing goes in exactly one category. Every vote is for one fruit, every bug is one kind of bug. If something could belong to two rows, or to none, the pictures would no longer add up to the number of things counted.
The lengths tell the truth only when the pictures are the same size and lined up. A row of bigger or spread-out pictures can look longer while holding fewer, as graph B shows.
A graph shows only what was counted. The fruit vote says which fruit these 20 students chose. It does not say why they chose it, or what another class would choose.
Picture graphs suit small counts. With one picture for each thing, a count of 45 needs a row of 45 pictures, which is slow to draw and slow to read. For counts like that, a bar graph (Making bar graphs) or a bigger key (Scaled picture graphs) does the job.

Where this goes next

A bar graph keeps the rows and drops the pictures: each row becomes one bar, and a numbered scale along its edge does the counting. Reading one is Reading a bar graph, and drawing one is Making bar graphs. A dot plot puts one mark for each thing above a number line, much like a picture graph whose categories are numbers (Reading dot plots: clusters, gaps and peaks, Making a dot plot).

The key grows up later. In Scaled picture graphs, one picture stands for several things, such as 2, 5 or 10, and half a picture stands for half as many. The habit started here, reading the key before counting anything, is what makes those graphs readable. Reading any graph someone else drew, starting from its title, labels and key, is Reading a graph: titles, labels and units.

Exercises

Day 1: read the key, then the rows

Exercises 1 to 4 use this graph of a fruit vote for a class picnic. Each circle stands for 1 student.

Figure 7The graph for Exercises 1 to 4 and 11 to 17.

How many students chose melons?
How many students chose kiwis?
Which fruit had the fewest votes, and how many?
Put the four fruits in order from fewest votes to most.
On a picture graph of bugs, the key says each diamond stands for 1 bug. The beetles row has 9 diamonds. How many beetles were counted?

Day 2: make a graph that tells the truth

Exercises 6 to 8 use this list. Zhuldyz looks under a log in the garden and writes down each bug she finds: ant, beetle, ant, cricket, ant, beetle, ant, ant, cricket, ant, beetle, ant, cricket. She makes a picture graph with rows for ants, beetles, crickets and moths.

How many pictures go in each row?
How many pictures does the whole graph have?
Which two rows are the same length?
A lost tooth tally shows January with one bundle of five and 2 loose marks, February with 4 loose marks, March with two bundles of five, and April with 1 loose mark. How many squares go in each column of the picture graph?
In another class, 23 children each vote once for one of four fruits. How many pictures does their picture graph have in all?

Day 3: questions the graph can answer

Exercises 11 to 17 use the picnic fruit graph.

How many more students chose kiwis than melons?
How many fewer students chose mangoes than cherries?
How many students voted in all?
How many students chose melons or cherries?
How many more students chose kiwis or cherries than chose melons or mangoes?
Three more students vote, and all three choose mangoes. How many students chose mangoes now? Which fruit does mangoes now tie with?
Before those late votes, how many more votes did mangoes need to tie with kiwis?
A lost tooth graph covers three months and shows 13 teeth in all. The September column has 4 squares and the October column has 6. How many squares are in the November column?
On a bug graph, the ants row has 9 diamonds. That is 4 more than the bees row. How many diamonds are in the bees row?

Explain

Why must every picture in a picture graph be the same size?
The key says each circle stands for 1 student. Everybody can see the circles, so why write the key at all?
A row has a category name but no pictures. What does it tell you, and why keep the row?
Is the row that looks longest always the one with the most? When can you trust the lengths?
Vladislava turns her fruit graph so that the rows become columns. Does any count change? Why or why not?

Answers

5 students. The melons row has 5 circles, and each circle is 1 student.
8 students. The kiwis row has 8 circles.
Mangoes, with 2 votes. Its row is the shortest, and it has 2 circles.
Mangoes 2, melons 5, cherries 6, kiwis 8. Each row is a little longer than the one before, and the counts agree: 2 < 5 < 6 < 8.
9 beetles. Each diamond is 1 bug, so 9 diamonds is 9 beetles.
Ants 7, beetles 3, crickets 3, moths 0. Cross out each word as it is counted; no moth is on the list, so the moths row is empty.
13 pictures. One picture for each bug, and the list has 7 + 3 + 3 + 0 = 13 bugs.
Beetles and crickets. Both have 3 pictures, and same-size pictures lined up from the same line make rows of the same length.
January 7, February 4, March 10, April 1. A bundle is 5, so 5 + 2 = 7 and 5 + 5 = 10.
23 pictures. Each child has exactly one picture, so the pictures match the children one to one.
3 more students. Kiwis has 8 and melons has 5, and 8 - 5 = 3. Lined up, 3 kiwi circles have no melon partner.
4 fewer students. Cherries has 6 and mangoes has 2: 6 - 2 = 4.
21 students. 8 + 2 = 10, 5 + 6 = 11, and 10 + 11 = 21.
11 students. 5 + 6 = 11.
7 more students. Kiwis or cherries: 8 + 6 = 14. Melons or mangoes: 5 + 2 = 7. And 14 - 7 = 7.
5 students, and it ties with melons. Mangoes had 2, and 2 + 3 = 5. Melons also has 5.
6 more votes. 2 + \square = 8, and counting on from 2 to 8 is 6 counts: 8 - 2 = 6.
3 squares. September and October make 4 + 6 = 10, and 10 + 3 = 13, so November has 13 - 10 = 3.
5 diamonds. The bees row is 4 shorter than 9: 9 - 4 = 5. Check: 5 + 4 = 9.
So that a longer row always means a bigger count. If one row’s pictures were bigger, that row could look longer while holding fewer pictures, and the graph would mislead anyone reading it at a glance.
The key tells the reader what a circle means, so they know that counting circles gives the number of students. Without it, a reader cannot be sure whether a circle is one student or several, and so cannot trust any count. Later graphs use keys where one picture stands for more than one thing, so reading the key first is a habit worth having now.
It tells you the count for that category is 0: it was counted, and nothing was found. Keeping the row shows the reader that the category was not forgotten.
No. Bigger or spread-out pictures can make a short count look long. The lengths can be trusted when every picture is the same size, the pictures are evenly spaced, and every row starts at the same line.
No count changes. Each student still has exactly one circle in the place for their fruit; only the direction the graph runs has changed, so counting each column gives the same numbers as counting each row did.