Grade 3 · 3.MD.B.3

Scaled picture graphs

Forty students voted for tacos. Do you really have to draw forty little pictures to show it?

You need

What multiplication meansGrade 2 Picture graphs and their keyGrade 2

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

Tahnee’s class ran a lunch vote, and 40 students chose tacos. A picture graph that draws one picture for each student would need a row of 40 plates for tacos alone. That row would be hard to draw, hard to count, and too long for the page.

A picture graph always comes with a key, the note that says how many things one picture stands for (Picture graphs and their key). So far the key has said “one picture, one thing”. A scaled picture graph changes the key: one plate stands for 10 students. Now the tacos row needs only 4 plates, because four groups of 10 make 40.

Reach for a scaled picture graph when the counts are too big to draw one by one, but you still want rows that can be compared at a glance. Reading one is multiplication. Each symbol is a group of the same size, and a row of symbols is equal groups, which is exactly what 4 \times 10 means (What multiplication means: 4 groups of 10).

The procedure

This lesson takes three days, one part a day. Part 1 reads a row. Part 2 asks questions that compare or combine rows. Part 3 turns a table of counts into a graph, which means choosing a key.

Part 1: reading a row

Read the key first. The key says what one whole symbol stands for. In the lunch vote below, one plate stands for 10 students. Every plate in the graph stands for the same 10; that is what makes the graph fair.

Figure 1Tahnee’s lunch vote. Each plate stands for 10 students, so half a plate stands for 5.

A row of whole symbols is equal groups. The tacos row has 4 plates, and each plate is a group of 10 students. Four groups of 10 is

4 \times 10 = 40

so 40 students chose tacos. Skip-counting along the row says the same thing, 10, 20, 30, 40, because 4 \times 10 means four tens. Notice that the number of plates, 4, is not the answer. It is the number of groups. The answer is how many students are in those groups.

Half a symbol stands for half the key. A plate stands for 10 students, so half a plate stands for half of 10, which is 5. The pasta row has 3 whole plates and half a plate:

\begin{aligned} 3 \times 10 &= 30 \\ 30 + 5 &= 35 \end{aligned}

so 35 students chose pasta. The whole plates are multiplied, and the half plate is added on. The same reasoning works for any key that splits into two equal whole halves: with a key of 4, half a symbol is 2; with a key of 2, half a symbol is 1.

Part 2: questions about the graph

Once each row has been turned into a count, the graph answers questions the way any set of counts does.

How many more, how many fewer. Subtract the smaller count from the larger one. That gap is the comparison taught in Subtraction as the difference: tacos has 40 and pasta has 35, so 5 more students chose tacos than pasta, and 5 fewer chose pasta than tacos. “More” and “fewer” ask about the same gap from the two ends.

There is also a shortcut through the symbols. Line the two rows up from the same starting edge. Where both rows have a symbol, those symbols stand for the same number of students, so they add nothing to the gap. Only the extra symbols at the end of the longer row make the difference. The tacos row has half a plate more than the pasta row, and half a plate is 5 students. Either way the answer is 5.

How many in all. Add the counts. All four rows of the lunch vote together are 40 + 35 + 20 + 15 students, worked out in Example 4.

Two steps. Some questions need a count that is not on the graph yet. “How many more students chose tacos or pasta than soup or salad?” first needs two totals, tacos and pasta together and soup and salad together, and only then a comparison. The first step builds the numbers the second step compares.

Part 3: drawing one from a table

To draw a scaled picture graph you start with a table of counts and choose the key yourself. A key works when every count in the table can be shown with whole symbols, or whole symbols and one half symbol.

Testing a key. Skip-count by the key toward each count. If you land exactly on the count, the row is all whole symbols. If you land half a key short of it, the row needs one half symbol at the end. If you land anywhere else, that key cannot show the count, because the leftover would be some other piece of a symbol, a third or a quarter of a picture, that nobody can read reliably.

For example, with a key of 4, the count 12 is 4, 8, 12: three whole symbols. The count 10 is 4, 8 and then 2 more, and 2 is half of 4: two whole symbols and a half. The count 9 is 4, 8 and then 1 more, and 1 is not half of 4, so a key of 4 cannot show 9.

Choosing among keys that work. A small key, such as 2, gives long rows. A bigger key gives shorter rows that fit the page, as long as it still passes the test for every count. Choose a key that passes for all the counts and keeps the longest row a comfortable length.

Drawing. Divide each count into groups of the key’s size to find how many symbols the row needs. Draw every symbol the same size and start every row at the same edge, so that a longer row really does mean a bigger count. A half symbol is drawn as half of a whole one, at the end of its row. Then write the key, so a reader can turn symbols back into counts. A full graph also needs a title and labels for its rows; those are the subject of Reading a graph: titles, labels and units.

Worked examples

Example 1

Reading the lunch vote

Use Tahnee’s lunch vote graph above, where each plate stands for 10 students. How many students chose each lunch?

Solution

Every whole plate is a group of 10, and half a plate is half of 10, which is 5.

Tacos has 4 whole plates: 4 \times 10 = 40 students.

Pasta has 3 whole plates and half a plate: 3 \times 10 = 30, and 30 + 5 = 35 students.

Soup has 2 whole plates: 2 \times 10 = 20 students.

Salad has 1 whole plate and half a plate: 1 \times 10 = 10, and 10 + 5 = 15 students.

Check the salad row by skip-counting: one plate is 10, and the half plate brings it to 15.

Example 2

Six and a half symbols

Koorine helps at the school library. Her graph of the week’s checkouts uses one book symbol for 4 books. How many books were checked out on Monday, and how many on Thursday?

Figure 2Library checkouts for one week. Each book symbol stands for 4 books.

Solution

Read the key first: one symbol is 4 books, so half a symbol is half of 4, which is 2 books.

Monday has 6 whole symbols and half a symbol. The whole symbols are 6 groups of 4:

6 \times 4 = 24

and the half symbol adds 2 more, so 24 + 2 = 26 books were checked out on Monday.

Thursday has 3 whole symbols and half a symbol: 3 \times 4 = 12, and 12 + 2 = 14 books.

The Monday row is 6 and a half symbols long, but the count is 26. Counting symbols alone would give a number four times too small, which is why the key is read before anything else.

Example 3

How many more, how many fewer

Use the same library graph. (a) How many more books were checked out on Friday than on Thursday? (b) How many fewer books were checked out on Tuesday than on Wednesday?

Solution

(a) Friday has 7 whole symbols, so 7 \times 4 = 28 books. Thursday has 14 books, from Example 2. The gap is

28 - 14 = 14

so 14 more books were checked out on Friday.

The symbol shortcut agrees. Line the rows up: Thursday’s 3 and a half symbols sit under Friday’s first 3 and a half, and those stand for the same books. Friday has 3 and a half symbols more. That is 3 \times 4 = 12 books, plus 2 for the half, so 14 books.

(b) Tuesday has 4 symbols, 4 \times 4 = 16 books. Wednesday has 5 symbols, 5 \times 4 = 20 books. The gap is 20 - 16 = 4, so 4 fewer books were checked out on Tuesday. The rows differ by exactly one whole symbol, and one symbol is 4 books, so this could be read straight off the graph.

Example 4

In all, and a two-step comparison

Use the lunch vote. (a) How many students voted in all? (b) How many more students chose tacos or pasta than chose soup or salad?

Solution

The counts from Example 1 are tacos 40, pasta 35, soup 20 and salad 15.

(a) Add all four. Add the two round numbers first, then the two others:

\begin{aligned} 40 + 20 &= 60 \\ 35 + 15 &= 50 \\ 60 + 50 &= 110 \end{aligned}

so 110 students voted. Adding in a different order is allowed because addition does not care about order (the commutative property) or grouping (the associative property).

(b) Neither count is on the graph yet, so the first step builds them.

Tacos or pasta: 40 + 35 = 75 students.

Soup or salad: 20 + 15 = 35 students.

The second step compares them: 75 - 35 = 40. So 40 more students chose tacos or pasta.

Check: 35 + 40 = 75, so the gap of 40 takes the smaller total up to the larger one.

Example 5

Late votes change a row

After the graph was drawn, 15 more students voted, and all of them chose soup. How many students chose soup now? How should the soup row change? Does soup now have more or fewer votes than tacos, and by how many?

Solution

The soup row had 2 plates, 20 students. Add the late votes: 20 + 15 = 35 students.

To draw 35 with plates of 10, skip-count: 10, 20, 30, then 5 more. Three whole plates make 30, and 5 is half of 10, so the row needs 3 whole plates and half a plate. It had 2 plates, so Tahnee adds 1 whole plate and half a plate. Check: the 15 new votes are one plate, 10, and half a plate, 5.

Tacos still has 40. Soup has 35, so soup has 40 - 35 = 5 fewer votes than tacos. On the graph the soup row is now half a plate shorter than the tacos row, and half a plate is 5 students.

Example 6

Choosing a key and drawing the graph

Koorine counts the books checked out from the library on one day, by kind:

Kind of book	Books checked out
Animals	12
Sports	8
Mystery	6
Space	10

Which of the keys 2, 4, 5 and 10 can show every count? Choose one and draw the graph.

Solution

Test each key by skip-counting toward every count.

Key 2. Every count is even, so skip-counting by 2 lands on each one exactly: 12 = 6 \times 2, 8 = 4 \times 2, 6 = 3 \times 2 and 10 = 5 \times 2. The rows are 6, 4, 3 and 5 symbols. This key works.

Key 4. 12 = 3 \times 4 and 8 = 2 \times 4 land exactly. For 6: 4, then 2 more, and 2 is half of 4, so one symbol and a half. For 10: 4, 8, then 2 more, so two symbols and a half. This key works too, with shorter rows.

Key 5. For 12: 5, 10, then 2 more. Half of 5 is not a whole number of books, and 2 is not a whole symbol either. This key fails.

Key 10. For 12: 10, then 2 more, and 2 is not half of 10. This key fails.

Both 2 and 4 work. A key of 4 keeps the longest row to 3 symbols, so Koorine chooses it. The rows are: animals 3 symbols, sports 2, mystery 1 and a half, space 2 and a half.

Figure 3The graph Koorine draws in Example 6, with each symbol standing for 4 books.

Check by reading the graph back: 1 \times 4 + 2 = 6 mystery books and 2 \times 4 + 2 = 10 space books, matching the table.

Example 7

The key is missing

Djalu finds a picture graph of a lunch vote with the key torn off. He knows from the class table that 24 students chose soup, and the soup row has 6 whole symbols. The rice row has 4 whole symbols and half a symbol. How many students does one symbol stand for, and how many chose rice?

Solution

The 24 soup votes are shared equally among 6 symbols, so each symbol stands for 24 \div 6 students (What division means). Since 6 \times 4 = 24,

24 \div 6 = 4

and one symbol stands for 4 students. Half a symbol stands for 2.

Rice: 4 \times 4 = 16 for the whole symbols, and 16 + 2 = 18 students.

Check the key on the soup row: 6 groups of 4 is 24, as the table says.

Boundaries

The key holds for the whole graph. Every whole symbol in one graph stands for the same number. Without a key, the rows still show which count is biggest, but not what any count is.

A half symbol needs a key that halves into whole things. Half of 10 is 5 and half of 4 is 2, but half of 5 students is not a number of students. A graph whose key is 5 (or any odd number) uses whole symbols only, so every count it shows lands exactly on the key: 5, 10, 15 and so on.

Only counts that fit can be shown. A key works for a count only when skip-counting lands on it or half a key short of it. A key of 10 cannot show 37; a different key, or a bar graph with its finer scale, is needed.

Compare counts, not symbols, across two graphs. Within one graph, more symbols means a bigger count. Two graphs with different keys cannot be compared symbol for symbol: 3 plates at 10 each is 30, but 5 symbols at 2 each is only 10.

The pictures must be the same size and lined up. A row is longer because it has more symbols, not because its symbols are drawn bigger. If one row’s pictures are wider or start further in, the lengths stop telling the truth about the counts.

Where this goes next

A scaled bar graph does the same job with a bar instead of a row of pictures: the numbers along its side play the part of the key, and Making bar graphs and Reading a bar graph read and draw one. Reading a graph someone else drew, starting from its title, labels and key, is Reading a graph: titles, labels and units. Comparisons such as “Room C checked out more than twice as many books as Room B” come later, in Times as many.

Exercises

Part 1: reading a row

Exercises 1 and 2 use this graph, where each book symbol stands for 5 books.

Figure 4The graph for Exercises 1, 2 and 7 to 11. Each symbol stands for 5 books.

How many books did Room C check out?
How many books did Room A check out?
A row on a library graph has 7 whole symbols, and each symbol stands for 2 books. How many books does the row show?
On a lunch vote graph, each plate stands for 10 students. The chili row has 5 whole plates and half a plate. How many students chose chili?
On a library graph, each symbol stands for 4 books. A row has 8 whole symbols and half a symbol. How many books does it show?
On a lunch vote graph, each plate stands for 10 students. The fish row has only half a plate. How many students chose fish?

Part 2: more, fewer and in all

Exercises 7 to 11 use the graph of books checked out by class.

How many more books did Room C check out than Room B?
How many fewer books did Room A check out than Room D?
How many books did the four classes check out in all?
Did Rooms A and B together check out more or fewer books than Room C? By how many?
Next week Room D checks out 15 more books than this week. How many symbols will its row have then?
On a graph of books checked out at the town library over one weekend, each symbol stands for 4 books. Saturday has 5 whole symbols and Sunday has 2 whole symbols and half a symbol. How many more books were checked out on Saturday?
On a lunch vote graph, each plate stands for 10 students. Pizza has 6 whole plates and half a plate, and rice has 3 whole plates. How many students chose pizza or rice in all?

Part 3: drawing one from a table

Exercises 14 and 15 use this table of the fruit students chose at lunch.

Fruit	Students
Apples	16
Pears	6
Plums	10
Peaches	14

With a key of 2 students for each symbol, how many symbols does each row need?
With a key of 4 students for each symbol, how many symbols does each row need?

Exercises 16 and 17 use this table from a lunch vote.

Lunch	Students
Noodles	40
Burgers	25
Chili	60
Salad	15

With a key of 10 students for each symbol, how many symbols does each row need?
Which counts in the table cannot be shown with a key of 4? Explain.
A row of 7 whole symbols stands for 35 books. What does one symbol stand for?
A row of 5 whole symbols and half a symbol stands for 22 students. What does one symbol stand for?
Djalu draws a graph with a key of 10 books. How many symbols does he draw for 70 books, and how many for 25 books?

Reasoning

Explain why a graph whose key is 5 students never uses half a symbol.
Two picture graphs use different keys. Is the row with more symbols always the bigger count? Give an example.
Explain why “how many more” can be found by counting the extra symbols in the longer row and using the key, without finding either row’s count.
A table is drawn once with a key of 2 and once with a key of 4. Explain why every row of the second graph has half as many symbols as the same row of the first.
Can a count of 13 be shown with a key of 4, using whole symbols and half symbols? Can it be shown with a key of 2? Explain.

Answers

45 books. Room C has 9 symbols, and 9 \times 5 = 45.
30 books. Room A has 6 symbols, and 6 \times 5 = 30.
14 books. 7 \times 2 = 14.
55 students. 5 \times 10 = 50, and half a plate is 5 more.
34 books. 8 \times 4 = 32, and half a symbol is 2 more.
5 students. Half a plate is half of 10.
25 more books. Room C has 45 and Room B has 4 \times 5 = 20, and 45 - 20 = 25. Room C’s row also has 5 extra symbols, and 5 \times 5 = 25.
5 fewer books. Room D has 7 \times 5 = 35 and Room A has 30, and 35 - 30 = 5: one symbol.
130 books. 30 + 20 + 45 + 35 = 130.
More, by 5 books. Rooms A and B together: 30 + 20 = 50. Room C: 45. And 50 - 45 = 5.
10 symbols. Room D will have 35 + 15 = 50 books, and 10 \times 5 = 50.
10 more books. Saturday: 5 \times 4 = 20. Sunday: 2 \times 4 + 2 = 10. And 20 - 10 = 10.
95 students. Pizza: 6 \times 10 + 5 = 65. Rice: 3 \times 10 = 30. And 65 + 30 = 95.
Apples 8, pears 3, plums 5, peaches 7. Each count is an even number: 16 = 8 \times 2, 6 = 3 \times 2, 10 = 5 \times 2, 14 = 7 \times 2.
Apples 4; pears 1 and a half; plums 2 and a half; peaches 3 and a half. 16 = 4 \times 4; 6 = 4 + 2; 10 = 8 + 2; 14 = 12 + 2, and 2 is half of 4.
Noodles 4; burgers 2 and a half; chili 6; salad 1 and a half. 25 = 20 + 5 and 15 = 10 + 5, and 5 is half of 10.
25 and 15. For 25: 6 \times 4 = 24 leaves 1. For 15: 3 \times 4 = 12 leaves 3. Neither leftover is 0 or 2, half of 4. The other two fit: 40 = 10 \times 4 and 60 = 15 \times 4.
5 books. 35 \div 7 = 5, since 7 \times 5 = 35.
4 students. 5 \times 4 = 20, and half a symbol is 2, so 20 + 2 = 22. A key of 2 gives 11 and a key of 10 gives 55, so 4 is the key.
7 symbols for 70 books, since 7 \times 10 = 70. For 25 books, 2 whole symbols and half a symbol, since 2 \times 10 = 20 and 5 is half of 10.
Half a symbol would stand for half of 5 students, and that is not a whole number of students. So every count on such a graph is a whole number of fives, and every row is whole symbols.
No. One graph with a key of 10 might show 2 plates, which is 20; another with a key of 2 might show 6 symbols, which is only 12. More symbols means more only inside one graph, where every symbol stands for the same number.
Line the rows up from the same edge. Where both rows have a symbol, the two symbols stand for the same number of things, so they add nothing to the gap. The gap is only the extra symbols, and the key says how many things they stand for: extra whole symbols times the key, plus half the key for an extra half symbol.
A symbol in the second graph stands for 4, which is two groups of 2. So every symbol in the second graph does the job of two symbols in the first, and a row needs half as many.
Not with a key of 4: skip-counting gives 4, 8, 12, and then 1 is left, which is neither a whole symbol nor half of one (2). With a key of 2 it can: 6 \times 2 = 12, and the 1 left is half of 2, so 6 symbols and a half.