Interpreting and Creating Scaled Picture Graphs (Pictographs) - Third Grade Mathematics

A picture graph (or pictograph) uses visual symbols or icons to represent data in an engaging, pictorial way. In earlier grades, each symbol on a picture graph stood for just 1 single item. In third grade, you take a big step forward into scaled picture graphs, where one symbol can represent 2, 5, 10, or more items! In this chapter, you will learn how to read keys, interpret full and half symbols, and create your own scaled pictographs.

The Secret of the Key

The most important part of any scaled picture graph is the key (sometimes called the legend). The key tells you the value of each symbol.

CRITICAL RULE: Always check the key before counting symbols!
If you do not read the key, you might count 4 symbols as 4 items,
when it really represents 4 x 5 = 20 items!

Example Key:
Each [ * ] = 2 books read.

Symbol Count:     What It Really Means:
[ * ]             2 books
[ * ] [ * ]       4 books
[ * ] [ * ] [ * ] 6 books

Reading Half Symbols

What happens when a data value is not a full multiple of the key? A scaled picture graph uses half of a symbol!

If 1 full star [ * ] = 4 students:
Then 1 half star [ ) ] = 2 students!  (Half of 4 is 2)

If a row has 3 full stars and 1 half star:
- 3 full stars: 3 x 4 = 12 students
- 1 half star:  2 students
- Total: 12 + 2 = 14 students!

Common Half Symbol Values:
Key: Full Symbol = 2   ->   Half Symbol = 1
Key: Full Symbol = 4   ->   Half Symbol = 2
Key: Full Symbol = 10  ->   Half Symbol = 5

Interpreting a Complete Scaled Picture Graph

Let us examine a complete scaled pictograph representing books read by four students during a reading challenge:

Books Read in March:
+-----------------------+---------------------------------------------+
| Student               | Books Read                                  |
+-----------------------+---------------------------------------------+
| Maya                  | [ # ] [ # ] [ # ] [ # ]                     |
| Liam                  | [ # ] [ # ]                                 |
| Sofia                 | [ # ] [ # ] [ # ] [ ) ]                     |
| Noah                  | [ # ] [ # ] [ # ] [ # ] [ # ]               |
+-----------------------+---------------------------------------------+
Key: Each [ # ] = 4 books. Each [ ) ] = 2 books.

Let us calculate each student's total books read:
- Maya: 4 full symbols -> 4 x 4 = 16 books.

- Liam: 2 full symbols -> 2 x 4 = 8 books.

- Sofia: 3 full symbols + 1 half symbol -> (3 x 4) + 2 = 12 + 2 = 14 books.

- Noah: 5 full symbols -> 5 x 4 = 20 books.


Now we can answer analysis questions:
- Who read the most books? Noah (20 books).

- Who read the fewest books? Liam (8 books).

- How many more books did Noah read than Maya? 20 - 16 = 4 books.

- How many books did all four students read combined? 16 + 8 + 14 + 20 = 58 books in all!


How to Create Your Own Scaled Picture Graph

To build a scaled picture graph from a frequency table:
1. Choose an appropriate scale: Look at your data values.
- If all numbers are even, a scale of 2 works well. - If numbers end in 0 or 5, a scale of 5 is ideal. - If numbers are large tens (20, 40, 50), a scale of 10 is perfect.
2. Clearly write the Key at the bottom of the graph.

3. Divide each category's value by the key number to find how many symbols to draw:
Number of Symbols = Category Total ÷ Key Value.
4. Draw full and half symbols neatly in rows.


Chapter Practice Exercises

  1. Look at this pictograph key: Each [ O ] represents 5 trees planted. Find the number of trees represented by: a. [ O ] [ O ] b. [ O ] [ O ] [ O ] [ O ] c. [ O ] [ O ] [ O ] and a half circle
  2. In a pictograph where each smiley face represents 10 students: a. How many smiley faces would you draw to show 40 students? b. How many smiley faces would you draw to show 25 students?
  3. A school store sold pencils: Monday: 12 pencils, Tuesday: 18 pencils, Wednesday: 6 pencils. a. If you make a pictograph with a key of 1 symbol = 6 pencils, how many symbols do you draw for each day? b. How many total pencils were sold across the three days?
  4. A pictograph shows 6 flower symbols for Roses and 3 flower symbols for Daisies. The key states that each flower symbol = 4 flowers. How many more roses are there than daisies?
  5. A student looks at a pictograph and says: "There are only 3 apples drawn for Class A and 5 apples drawn for Class B, so Class B has 2 more apples." Explain what vital piece of information the student forgot to check.

Solutions and Step-by-Step Answers

  1. Tree calculations (Key = 5 trees): a. 2 symbols: 2 x 5 = 10 trees. b. 4 symbols: 4 x 5 = 20 trees. c. 3 full symbols (15) + half symbol (2.5 -> or if discrete, 5 / 2 = 2 or 3; usually keys with even values use halves, for key of 5, a half represents 2.5). 3 x 5 = 15 trees plus 2.5 = 17.5 trees.
  2. Smiley face graph (Key = 10 students): a. For 40 students: 40 ÷ 10 = 4 full smiley faces. b. For 25 students: 2 full smiley faces (20) and 1 half smiley face (5) = 2 and a half smiley faces.
  3. School store pencils (Key = 6 pencils): a. Monday: 12 ÷ 6 = 2 symbols. Tuesday: 18 ÷ 6 = 3 symbols. Wednesday: 6 ÷ 6 = 1 symbol. b. Total pencils: 12 + 18 + 6 = 36 pencils (or 6 symbols x 6 = 36).
  4. Roses vs. Daisies:
  5. Roses: 6 x 4 = 24 roses.
  6. Daisies: 3 x 4 = 12 daisies. Difference: 24 - 12 = 12 more roses (or 3 extra symbols x 4 = 12).
  7. The student forgot to check the Key! If each apple symbol represents 10 real apples, Class B actually has 20 more apples, not just 2. You must multiply the symbol count by the scale value in the key.