How to teach expanded notation with powers of 10
Year 7 (ages 12 to 13)
Expanded notation writes a number as the sum of each digit multiplied by its place value, expressed as a power of 10 (e.g. 4 x 10^3). It makes the value of each digit explicit and connects place value to exponent notation.
How to teach it
- Review place value first: in 4,502, the 4 is worth 4,000, the 5 is worth 500, and so on.
- Introduce powers of 10 as a shorthand for place value: 1,000 = 10^3, 100 = 10^2, 10 = 10^1, 1 = 10^0.
- Write a number in expanded form by pairing each nonzero digit with its power of 10, then adding the terms.
- Practise the reverse: given an expanded form, add the terms to find the original number.
- Extend to larger numbers (millions, billions) to reinforce that the pattern of powers keeps working at any scale.
Worked example
Write 30,406 in expanded form 3 x 10^4 + 0 x 10^3 + 4 x 10^2 + 0 x 10^1 + 6 x 10^0 Skip the zero terms: 3 x 10^4 + 4 x 10^2 + 6
Common mistakes
- Using the wrong power of 10 for a digit's position, especially miscounting for larger numbers.
- Including a term for a zero digit instead of leaving it out.
- Forgetting that the ones digit pairs with 10^0 (which equals 1), not 10^1.
- Adding the digits themselves instead of each digit's full place value when reversing the process.
Frequently asked questions
What is expanded form using powers of 10?
Expanded form breaks a number into the value of each of its digits, written as that digit multiplied by a power of 10. For example, 4,502 = 4 x 10^3 + 5 x 10^2 + 0 x 10^1 + 2 x 10^0.
Why use powers of 10 instead of just writing place-value names (thousands, hundreds...)?
Powers of 10 notation (10^3, 10^2...) generalises cleanly to any size of number and connects directly to exponent notation used elsewhere in secondary maths, rather than relying on a growing list of place-value names.
Do you include zero digits in expanded form?
No, a zero digit contributes nothing (anything times zero is zero), so those terms are simply left out of the expanded sum.
What year is expanded notation with powers of 10 taught?
In the Australian Curriculum this is a Year 7 skill (AC9M7N03), representing whole numbers in expanded form using place value and powers of ten, building on the F-6 idea of expanded form with an explicit exponent notation.
Practise with free worksheets
Printable worksheets with answer keys that are never wrong.