Place Value Models up to 999

A place value model uses physical or drawn shapes to show how much a number is really worth. For numbers up to 999, there are three shapes to know:

• A hundreds flat is a large square block worth 100.
• A tens rod is a long thin block worth 10.
• A ones cube is a small square block worth 1.

Since every number up to 999 has at most three digits, at most nine hundreds flats, nine tens rods, and nine ones cubes are ever needed to build it. Counting how many of each block a model uses is the first step to reading its value.

To read a model, count each group of blocks by itself, starting with the biggest block.

Count the flats first: there are 3, so that group is worth 300. Next count the rods: there are 4, so that group is worth 40. Finally count the cubes: there are 2, so that group is worth 2. Adding the groups gives the number the model represents:

\( 300 + 40 + 2 = 342 \)

This is the same idea behind identifying hundreds, tens, and ones in a written number, except here the digits are shown as blocks instead of numerals.

The reverse skill is just as important: given a number, decide how many of each block to use. For a number written as a three digit value with hundreds digit \( a \), tens digit \( b \), and ones digit \( c \), the model always follows the pattern

\( 100a + 10b + c \)

For example, to model 507, use 5 hundreds flats, 0 tens rods, and 7 ones cubes. Notice that a digit of 0 simply means that group of blocks is left out of the model entirely, not that a special "zero block" is drawn.

Suppose a model shows 6 hundreds flats, 1 tens rod, and 8 ones cubes. What number does it represent?

Add the value of each group: \( 6 \times 100 = 600 \), \( 1 \times 10 = 10 \), and \( 8 \times 1 = 8 \). Adding these gives \( 600 + 10 + 8 = 618 \).

Now try it the other way: model the number 429. This needs 4 hundreds flats, 2 tens rods, and 9 ones cubes, since \( 4 \times 100 + 2 \times 10 + 9 \times 1 = 429 \).

An interesting feature of base ten blocks is that the same number can sometimes be shown with more than one combination of blocks. For example, 10 ones cubes can always be traded for 1 tens rod, and 10 tens rods can always be traded for 1 hundreds flat, without changing the total value. This trading idea is the foundation for regrouping place values up to 999, which becomes especially useful once addition and subtraction with carrying and borrowing are introduced.

• Mixing up which block is worth 100 and which is worth 10. Remember: the flat is the biggest and flattest shape, the rod is long and thin, and the cube is the smallest.
• Forgetting that a missing group (zero flats, zero rods, or zero cubes) still counts as a digit of 0 in the number, it is not skipped when writing the number down.
• Adding up the number of blocks instead of their total value, for example treating 3 flats and 2 rods as "5" instead of 320.

Once these block models feel comfortable, the next step is usually writing the same value in expanded form up to 999, and practicing matching place value models to numbers to build speed and confidence.

Related lessons

• Math 2