Place Value Models of Multi-Digit Numbers

A place value model is a picture built from base ten blocks that shows exactly how many ones, tens, hundreds, and thousands are inside a multi-digit number. Instead of just seeing the digits \(2,431\), you see the number broken into groups you can count, stack, and compare. This is one of the most useful ways to understand place value before moving on to skills like identifying the place value of a digit or writing numbers in expanded form.

Base ten materials use four standard shapes, and each one stands for a different place value:

• A small cube (a "one") stands for 1.
• A rod made of 10 small cubes in a row (a "ten") stands for 10.
• A flat made of 10 rods side by side (a "hundred") stands for 100.
• A large cube made of 10 flats stacked up (a "thousand") stands for 1,000.

Every block is exactly 10 times bigger than the block before it. This is the same "groups of 10" idea you use when you convert between place values, just shown as a picture instead of numbers.

To model a number like \(2,431\), sort the digits by place value first, then choose the matching blocks:

• The digit 2 is in the thousands place, so use 2 large cubes.
• The digit 4 is in the hundreds place, so use 4 flats.
• The digit 3 is in the tens place, so use 3 rods.
• The digit 1 is in the ones place, so use 1 small cube.

The picture below shows this model. Notice how the blocks get smaller as you move from thousands down to ones, matching the way place values shrink from left to right in the written number.

The most important idea behind these models is regrouping: 10 of any block always trades in for 1 of the next block up. Ten small cubes line up to form exactly 1 rod, ten rods line up to form exactly 1 flat, and ten flats stack to form exactly 1 large cube. The diagram below shows a flat built from 10 rods, which is why 10 tens equal 1 hundred.

Suppose a picture shows 3 large cubes, 6 flats, 0 rods, and 5 small cubes. What number does this model?

Count each group by its place value: 3 thousands is \(3 \times 1000 = 3000\), 6 hundreds is \(6 \times 100 = 600\), 0 tens is \(0\), and 5 ones is \(5\). Adding the groups gives \(3000 + 600 + 0 + 5 = 3605\). Since there are no rods, the tens digit is written as 0 to hold that place, so the number is \(3,605\).

Once you can build and read a model like this, it becomes much easier to organize the same information in a place value table, where each block group is written under its own column.

Place value models make an abstract idea, that each digit's value depends on its position, into something you can physically see and touch. Before working only with digits, being able to build a number out of ones, tens, hundreds, and thousands blocks gives a strong foundation for comparing numbers, adding and subtracting with regrouping, and understanding why our number system is built on groups of 10.

Related lessons

• Math 3