What is ‘expanded form’ in math?
Never heard of expanded form? Fear not. This post will walk you through it, step by step. Prepare to be baffled no longer.
Does your child’s math homework involve questions about something mysterious called ‘expanded form‘?
Never heard of it?
Don’t worry, you’re certainly not alone.
This post will explain what expanded form is and walk you through some examples.
Be mystified no longer and read on!
Expanded form and standard form
Expanded form is a term that crops up when we are looking at the math topic of place value.
You’ll often see questions that ask you to write a number in expanded form. Or, perhaps to convert a number written in expanded form back into its standard form.
Standard form refers to numbers in the form that we’re accustomed to seeing them, such as 356, 34 or 4.67.
When we write a number in expanded form, we are essentially stretching that number out to show the value of each of its digits.
Let’s look at some examples.
How to write numbers in expanded form, with examples
Example 1
Let’s take the 3-digit number 542.
If we break 542 down into its different place value columns we can see that it has a 5 in the hundreds column, a 4 in the tens column and a 2 in the ones column:
With expanded form, we can take this one step further and show the value of each of those digits as numbers.
In expanded form, 542 is written as:
The value of the 5 in the hundreds column is 500 (5 x 100), the value of the 4 in the tens column is 40 (4 x 10) and the value of the 2 in the ones column is 2 (2 x 1).
The value of each digit is connected to the next with a plus sign.
Once you see a number written in expanded form, you can see where the name comes from: the number is expanded/stretched to show the value of each of its digits.
Example 2
This time let’s take a number that has 4 digits, such as 9831.
First let’s write the number so that each digit sits under the correct place value headings. As we have a 4-digit number this time, we’ll need to include the thousands column as well.
So, if we line up this number under the correct place value headings we can see that we have a 9 in the thousands column, an 8 in the hundreds column, a 3 in the tens column and a 1 in the ones column:
Now let’s look at 9831 in expanded form:
The value of the 9 in the thousands column is 9000 (9 x 1000), the value of the 8 in the hundreds column is 800 (8 x 100), the value of the 3 in the tens column is 30 (3 x 10) and the value of the 1 in the ones column is 1 (1 x 1).
What about numbers where one of the digits is a zero?
Let’s look at an example where one of the digits is a zero, for example 7204.
To start with, let’s take a quick look at the place value columns to see where each digit sits.
We have 7 in the thousands column, a 2 in the hundreds column, a 0 in the tens column, and a 4 in the ones column.
If we ‘expand’ 7204, it would look like this:
The value of the 7 in the thousands column is 7000 (7 x 1000), the value of the 2 in the hundreds column is 200 (2 x 100), the value of the 4 in the ones column is 4 (4 x 1).
As you can see, we don’t need to include a value for the tens digit in the expanded form, because the tens digit was 0 (0 x 10), so it has no value.
Even though we don’t show the value of the 0 in expanded form, in standard form the 0 is crucial because it is a place holder. If we took the 0 out of 7204 in standard form, it would become 724 and a different number entirely.
Related post: What is a place value chart? (including free printable charts)
Expanded form with decimals
We can also show a decimal number in expanded form. It works in exactly the same way, except we will be looking at different place value columns this time.
Let’s take 3.586 as an example.
We will line the decimal number under the correct place value headings.
This time, we have a 3 in the ones column, a 5 in the tenths column, a 6 in the hundredths column and an 8 in the thousandths column.
Just as with our previous examples, we will write the the value of each digit in the number, connecting each one with a plus sign.
We start from the left with the digit that has the greatest value, in this case the ones digit.
So, 3.568 in expanded form would look like this:
The value of the 3 in the ones column is 3 (3 x 1), the value of the 5 in the tenths column is 0.5 (5 x 0.1), the value of the 6 in the hundredths column is 0.06 (6 x 0.01) and the value of the 8 in the thousandths column is 0.008 (8 x 0.001).
You can also write this number in expanded form with the tenths, hundredths and thousandths shown as fractions like so:
Expanded form vs Expanded notation
You may also come across a term called expanded notation.
Although similar, there is a slight difference between expanded form and expanded notation.
As we’ve seen, with expanded form, a number is shown as the sum of the value of each of its digits:
With expanded notation, a number is instead shown as the sum of each digit multiplied by its place value (in this example by 1, 10 or 100):
Essentially, expanded notation shows the math calculations we make in order to reach expanded form.
Why is expanded form useful?
When you’re learning about place value and working with large numbers, expanded form is really helpful. To be able to write a number in expanded form, you have to really understand how the place value system works and know what the value of each digit in a number is.
Once you’re able to partition a number into its different place value components, you can use this to help you with other calculations.
For example, let’s take addition. Say you needed to calculate 48 + 33 in your head.
If you can break each number down into its tens and ones (just like you would if you were converting those numbers into expanded form), it makes the calculation much easier:
You would add up the ones (11), add up the tens (70), and then total those two numbers for your answer (81).
And there it is. A quick crash-course in expanded form so you’re up to speed for your child’s next piece of homework. I hope this was helpful. Thanks for reading.
More place value posts from Math, Kids and Chaos:
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