Picture Graphs with Multi-Unit Scales (2, 5, 10) - Second Grade Mathematics

When data sets become large—such as surveying 50 students, counting 80 library books, or tracking 100 cans collected for a food drive—drawing a single picture for every single item becomes impossible; your paper would run out of room and your hand would cramp! To represent large quantities neatly and efficiently, mathematicians use scaled picture graphs, where each symbol represents a multi-unit scale of 2, 5, or 10 items. In this chapter, you will master reading scaled picture graphs by skip-counting symbols, learn how to interpret half-symbols, and construct your own scaled graphs.

Multi-Unit Keys: Symbols Worth More Than One

In a scaled picture graph, the key unlocks the true value of each symbol through skip-counting.

Scaled Graph: Cans Collected for Food Drive
+-------------------------------------------------------+
|  CANS COLLECTED BY CLASS                              |
+-------------------------------------------------------+
|  Class A   |  [#]  [#]  [#]                           |
|  Class B   |  [#]  [#]  [#]  [#]  [#]                 |
|  Class C   |  [#]  [#]                                |
|  Class D   |  [#]  [#]  [#]  [#]                      |
+-------------------------------------------------------+
|  KEY: Each [#] represents 10 cans                     |
+-------------------------------------------------------+

Skip-Counting by the Key Scale

To find how many cans Class B collected, DO NOT count: "1, 2, 3, 4, 5 cans." Look at the key: Each [#] = 10 cans! Skip-count by tens: 10, 20, 30, 40, 50 cans! Class A: 3 symbols x 10 = 30 cans. Class B: 5 symbols x 10 = 50 cans. Class C: 2 symbols x 10 = 20 cans. Class D: 4 symbols x 10 = 40 cans. The scaled key compressed 140 cans of food into just 14 neat symbols!

Common Scale Factors: 2s, 5s, and 10s

Second-grade scaled graphs focus on the three friendliest skip-counting patterns.

The Three Classic Scale Factors:

Scale of 2:  Each symbol = 2 items.
             Skip-count by 2s: 2, 4, 6, 8, 10...
             4 symbols = 4 x 2 = 8 items.

Scale of 5:  Each symbol = 5 items.
             Skip-count by 5s: 5, 10, 15, 20, 25...
             4 symbols = 4 x 5 = 20 items.

Scale of 10: Each symbol = 10 items.
             Skip-count by 10s: 10, 20, 30, 40, 50...
             4 symbols = 4 x 10 = 40 items.

Connecting to Multiplication Foundations

Notice how scaled graphs directly apply your equal groups understanding: 3 symbols with a key of 5 is literally "3 groups of 5" (3 x 5 = 15). 6 symbols with a key of 2 is literally "6 groups of 2" (6 x 2 = 12). Reading scaled graphs strengthens your multiplication fluency every time you count.

The Half-Symbol: Cutting a Key in Half

What happens if a category has an odd number of items, or a quantity that lands between multiples of the scale? Mathematicians draw half of a symbol!

Interpreting Half-Symbols:

When Key is: Each (O) = 2 items:
             Half a symbol (|) = 1 item (half of 2).
             Example: (O) (O) (|) = 2 + 2 + 1 = 5 items.

When Key is: Each [#] = 10 items:
             Half a symbol [| = 5 items (half of 10).
             Example: [#] [#] [#] [| = 10 + 10 + 10 + 5 = 35 items.

The Value of Half

A half-symbol represents exactly half the value of the key: If the whole symbol is worth 2, half the symbol is worth 1. If the whole symbol is worth 10, half the symbol is worth 5. When you see a half-symbol at the end of a row, count all the full symbols first, then add the half-value to finish your total.

Practice Exercises

Exercise 1: A picture graph titled "Trees Planted in the Park" has a Key: Each [Tree] = 5 trees. Oak has 6 symbols, Pine has 4 symbols, and Maple has 7 symbols. Calculate the number of each type of tree planted.

Exercise 2: In a graph where each smiley face represents 2 children, a row shows 5 full smiley faces and 1 half smiley face. How many children are represented in that row? Show your addition.

Exercise 3: In a graph where each box represents 10 books, how many whole boxes and half boxes would you draw to represent 45 books?

Exercise 4: Explain why choosing a scale of 10 is better than a scale of 1 when displaying data for a school fundraiser that collected 350 pennies.

Solutions and Explanations

Solution to Exercise 1: Use the key (each symbol = 5 trees) and skip-count by fives: Oak: 6 x 5 = 30 oak trees. Pine: 4 x 5 = 20 pine trees. Maple: 7 x 5 = 35 maple trees.

Solution to Exercise 2: Each full smiley face is worth 2 children, and half a smiley face is worth 1 child (half of 2). Five full symbols equal 5 x 2 = 10 children. Add the half symbol: 10 + 1 = 11 children.

Solution to Exercise 3: To represent 45 books with a scale of 10: Each whole box is 10 books, so 4 whole boxes represent 40 books (4 x 10 = 40). The remaining 5 books is half of 10, represented by 1 half box. You would draw 4 whole boxes and 1 half box.

Solution to Exercise 4: If you used a scale of 1, you would have to draw 350 individual pictures on your paper! That would take hours, take up massive amounts of space, and be almost impossible for a reader to count without losing track. Using a scale of 10 allows you to represent all 350 pennies using just 35 symbols (or a scale of 50 using 7 symbols), making the graph compact, tidy, and easy to read.