I’m trying to clean up my understanding of place value when there are zeros in the middle of a number. I keep feeling confident, then I trip over expanded form or regrouping and second-guess myself. I remember losing points on a quiz because I “simplified” an expanded form in a way my teacher said wasn’t valid, and I’ve been cautious ever since.
Example: with 3,040, I label it as 3 thousands, 0 hundreds, 4 tens, 0 ones. In expanded form I wrote 3,000 + 40. Then I tried regrouping and wrote 30 hundreds + 40, and also 304 tens + 0 ones. Is it okay to say 304 × 10, or is that mixing forms in a way that’s not standard?
Another example: 40,506. I wrote 4 ten-thousands, 0 thousands, 5 hundreds, 0 tens, 6 ones. For expanded form I put 40,000 + 500 + 6, but I also tried two “regrouped” versions: 405 × 100 + 6 and 40 × 1,000 + 506. Are either of those considered correct ways to show regrouped place value, or am I breaking a rule by compressing across a zero place?
More generally, what’s the correct way to handle zeros in expanded form? Do I need to include terms like 0 × 1,000 or 0 × 10, or is leaving them out better? And when I regroup (like trading 1 thousand for 10 hundreds), is there a clean, consistent rule so I don’t accidentally change the value or the form? I’d really appreciate a straightforward way to check myself on numbers like 3,040 and 40,506 because I keep making small mistakes around the zeros.
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3 Responses
Zeros just mean “skip that place” in expanded form-so 3,040 = 3,000 + 40 and 40,506 = 40,000 + 500 + 6; regrouping is fine too (30 hundreds + 40, or 304 tens = 304×10), and 40×1,000 + 506, though 405×100 + 6 isn’t standard since you shouldn’t jump past the thousands place.
I once lost points for “helpfully” writing every 0×place term, so now I only include zeros if the teacher wants multiplicative form and I sanity-check by trading 1 of a unit for 10 of the next (good refresher: https://www.khanacademy.org/math/cc-fourth-grade-math/imp-place-value-and-rounding/imp-expanded-form/a/expanded-form).
You’re not overthinking-zeros just mean “no of that unit,” so in expanded form you can include 0×(place) or leave it out, and in regrouping you can convert everything to one unit as long as the value stays equal, so 304×10, 30×100+40, 405×100+6, and 40×1000+506 are all valid.
Example: 3,040 = 3×1000 + 0×100 + 4×10 + 0×1 = 3,000 + 40 = 30×100 + 40 = 304×10; and 40,506 = 4×10,000 + 0×1000 + 5×100 + 0×10 + 6 = 40,000 + 500 + 6 = 405×100 + 6 = 40×1000 + 506.
Zeros are the quiet seat-savers of place value: they hold the place, but they add nothing. So in standard expanded form you write a sum of digit × place-value units, including zero terms only if asked (3×1000 + 0×100 + 4×10 + 0×1), and it’s perfectly normal to drop the zero terms and just write 3,000 + 40. Regrouping (renaming) is a different game: you can trade 1 of a larger unit for 10 of the next smaller, as long as you keep the units clear and stick to one style on a line. For 3,040, these are all fine: 30 hundreds + 4 tens, 304 tens, or 304 × 10; the last two are “product” or “unit” forms, not expanded form, but they’re correct renamings. For 40,506, your 40,000 + 500 + 6 is standard expanded form, and both 405 × 100 + 6 and 40 × 1,000 + 506 are valid regroupings-even though you “jump” over zero places, that’s okay because zeros contribute nothing to the sum. A tidy self-check: (1) if you’re doing expanded form, make it a sum of digit × place units; (2) if you’re regrouping, either express everything in one unit (when it divides evenly) or as “n units of size U plus a remainder smaller than U”; and (3) recombine to see if you land back on the original number. I’m 99% sure most teachers would mark your versions as right when labeled clearly, but some are picky about the exact form, so match their directions. Nice refresher here: https://www.mathsisfun.com/numbers/expanded-form.html